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)

We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$. In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$.

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)

There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions (...) of weakness. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems. (shrink)

The paper pursues a premise semantics for counterfactuals. The way conflicts are resolved in a premise semantics depends on the way the premises are divided up and lumped together. Are there deep and non-trivial principles guiding this process that might be worth exploring? The article explores a positive answer to this question, which is then taken up in much more detail in my 'An investigation of the lumps of thought', in particular in the 2012 version. -/- The article has been (...) reprinted without substantial changes in my 2012 'Modals and Conditionals. New and Revised Perspectives.'. (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)

A theorem on the partitioning of a randomly selected large population into stationary and non-stationary components by using a property of the stationary population identity is stated and proved. The methods of partitioning demonstrated are original and these are helpful in real-world situations where age-wise data is available. Applications of this theorem for practical purposes are summarized at the end.

For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely $\kappa $ Borel sets? Work of Lusin, Souslin, and Hausdorff shows that ${\mathbb R}$ can be partitioned into $\aleph _1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of ${\mathbb R}$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega $ with $0,1 \in A$, there is a forcing (...) extension in which ${A = \{ n :\, \text {there is a partition of } {{\mathbb R}} \text { into }\aleph _n\text { Borel sets}\}}$. We also look at the corresponding question for partitions of ${\mathbb R}$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa $ such that there is a partition of ${\mathbb R}$ into precisely $\kappa $ closed sets can be fairly arbitrary. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the vector (Hilbert) space version of the mathematics of partitions at the set level. Since partitions are the math tool to describe indefiniteness and definiteness, this shows how the reality so well described by QM is a non-classical reality featuring the objective indefiniteness of superposition states. The lattice of partitions gives a skeletal model of quantum reality with the partition versions of pure states, (...) non-classical mixed states (i.e., including a superposition state), and the completely distinguished classical state that satisfies the partition logic version of the Principle of Identity of Indistinguishables. Both the key notions of quantum states and quantum observables are respectively the (density) matrix versions of partitions and vector space version of a numerical attributes. The operation of projective measurement given by the Lüders mixture operation is the vector space version of the partition join operation between the partition prefiguring the density matrix of the state being measured and the partition prefiguring the observable being measured. All this adds specific key concepts and structure to Abner Shimony’s idea of a literal understanding of QM to form what might be called the "Objective Indefiniteness Interpretation" of QM. (shrink)

This book presents a new ‘partitional' approach to understanding or interpreting the math of standard quantum mechanics (QM). The thesis is that the mathematics (not the physics) of QM is the Hilbert space version of the math of partitions on a set and, conversely, the math of partitions is a skeletonized set level version of the math of QM. Since at the set level, partitions are the mathematical tool to represent distinctions and indistinctions (or definiteness and indefiniteness), this approach shows (...) how to interpret the key non-classical QM notion of superposition in terms of (objective) indefiniteness between definite alternatives (as opposed to seeing it as the sum of ‘waves'). Thus, the book develops a new mathematical, or indeed, logical, approach to the century-old problem of interpreting quantum mechanics, ensure it is of interest to philosophers of science as well as mathematicians and physicists. (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)

Nineteenth and twentieth century philosophies of science have consistently failed to identify any rational basis for the compelling character of scientific analogies. This failure is particularly worrisome in light of the fact that the development and diffusion of certain scientific analogies, e.g. Darwin’s analogy between domestic breeds and naturally occurring species, constitute paradigm cases of good science. It is argued that the interactivist model, through the notion of a partition epistemology, provides a way to understand the persuasive character of (...) compelling scientific analogies without consigning them to an irrational or arational context of discovery. (shrink)

Two partition-theorems are proved for a particular causal decision theory. One is restricted to a certain kind of partition of circumstances, and analyzes the utility of an option in terms of its utilities in conjunction with circumstances in this partition. The other analyzes an option's utility in terms of its utilities conditional on circumstances and is quite unrestricted. While the first form seems more useful for applications, the second form may be of theoretical importance in foundational exercises. (...) Comparisons are made with theorems of Richard Jeffrey, Brad Armendt, and Peter Fishburn. (shrink)

For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated . It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ ≥ (...) K. For strongly inaccessible K we show that K is K-almost compact iff K is weakly compact, and if K is 2K-almost compact, then K is measurable. Further K is strongly compact iff it is γ-almost compact for all γ ≥ K. (shrink)

We consider colorings of the pairs of a family \ of topological type \, for \; and we find a homogeneous family \ for each coloring. As a consequence, we complete our study of the partition relation \^2_{l,m}}\) identifying \ as the smallest ordinal space \^2_{l,4}}\).

It is shown that the notion of the partition of a set can be used to describe in a uniform way the meaning of the expression the same, in its basic uses in transitive and ditransitive sentences. Some formal properties of the function denoted by the same, which follow from such a description are indicated. These properties indicate similarities and differences between functions denoted by the same and generalised quantifiers.

Lewis sender‐receiver games illustrate how a meaningful term language might evolve from initially meaningless random signals (Lewis 1969; Skyrms 2006). Here we consider how a meaningful language with a primitive grammar might evolve in a somewhat more subtle sort of game. The evolution of such a language involves the co‐evolution of partitions of the physical world into what may seem, at least from the perspective of someone using the language, to correspond to canonical natural kinds. While the evolved language may (...) allow for the sort of precise representation that is required for successful coordinated action and prediction, the apparent natural kinds reflected in its structure may be purely conventional. This has both positive and negative implications for the limits of naturalized metaphysics. (shrink)

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

We study how to partition a set of agents in a stable way when each coalition in the partition has to share a unit of a perfectly divisible good, and each agent has symmetric single-peaked preferences on the unit interval of his potential shares. A rule on the set of preference profiles consists of a partition function and a solution. Given a preference profile, a partition is selected and as many units of the good as the (...) number of coalitions in the partition are allocated, where each unit is shared among all agents belonging to the same coalition according to the solution. A rule is stable at a preference profile if no agent strictly prefers to leave his coalition to join another coalition and all members of the receiving coalition want to admit him. We show that the proportional solution and all sequential dictator solutions admit stable partition functions. We also show that stability is a strong requirement that becomes easily incompatible with other desirable properties like efficiency, strategy-proofness, anonymity, and non-envyness. (shrink)

We consider finite partitions of the closure F¯ of an ωα-uniform barrier F. For each partition, we get a homogeneous set having both the same combinatorial and topological structure as F¯, seen as a subspace of the Cantor space 2N.

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)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on Pκ that satisfy the square brackets partition relation equation image, where μ is a cardinal ≥2.

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)

Dans le contexte de la pandémie de COVID-19, les pratiques décisionnelles liées à la répartition des ressources médicales et au traitement des personnes âgées nous renseignent sur les éthiques présentes en milieu de soin et au niveau sociétal. La comparaison entre la prise de décision dans le contexte quotidien et les particularités d’une éthique de pandémie met en lumière les tenants du passage entre une éthique hors pandémie et une « pandéthique ». L’approche éthique de santé publique, notamment utilitariste, a (...) été mise de l’avant d’une manière prépondérante dans les débats et dilemmes éthiques entourant l’allocation des ressources et la priorisation. En soulevant les oppositions et enjeux associés aux discours et aux choix du rationnement en fonction de l’âge émerge la question du traitement des personnes âgées en contexte de COVID-19, et de l’âgisme vécu dans ce contexte. En parallèle, les décisions et choix éthiques difficiles s’entremêlent au duty to care du soignant, et par conséquent à la possibilité de blessure morale. Un conflit émerge entre les pratiques éthiques décisionnelles et les valeurs personnelles ou professionnelles du soignant, alors que l’équilibre entre ses divers devoirs est bouleversé. Des approches et éthiques alternatives sont ainsi mises de l’avant à la lumière des situations vécues, notamment en contexte de soin de longue durée. La thèse développée vise à soutenir la valeur ajoutée de l’anthropologie aux processus décisionnels et son intégration plus formelle aux approches bien connues en bioéthique. À partir du regard anthropologique, nous ouvrons en conclusion sur des pistes de réflexion associées aux éthiques de la discussion, de la vulnérabilité, féministes, ou encore du care comme d’autres manières d’aborder la prise de décision en contexte de pandémie, à un moment où la réflexion éthique et sociale s’impose comme capitale. (shrink)

It is known that in an infinite very weakly cancellative semigroup with size kappa $\kappa $ κ, any central set can be partitioned into kappa $\kappa $ κ central sets. Furthermore, if kappa $\kappa $ κ contains lamda $\lambda $ λ almost disjoint sets, then any central set contains lamda $\lambda $ λ almost disjoint central sets. Similar results hold for thick sets, very thick sets and piecewise syndetic sets. In this article, we investigate three other notions of largeness: quasi-central (...) sets, C-sets, and J-sets. We obtain that the statement applies for quasi-central sets. If the semigroup is commutative, then the statement holds for C-sets. Moreover, if kappa Superscript omega Baseline equals kappa $\kappa ^\omega = \kappa $ κω=κ, then the statement holds for J-sets. (shrink)

There are fundamentally two mathematical logics. One the Boolean logic of subsets, usually presented today in the special case of propositional logic, which has many sublogics and extensions, the most important being the intuitionistic logic usually modeled by the open subsets of a topological space. The other co-fundamental mathematical logic is the topic of this book, the logic of partitions. We are using ”logic” in a mathematical sense as being about basic mathematical objects, subsets of a universe set or partitions (...) on a universe set.1 Logic, in this mathematical sense, is not about propositions, although, as with any mathematical theory, it involves propositions about the basic objects, e.g., that an element is in a subset or that a distinction is made by a partition. Moreover, by taking the universe set to be the one element set 1, with two subsets ∅ and 1, there is a special case of subset logic, namely propositional logic, that can be interpreted as being about propositions with ∅ representing falsehood and 1 representing truth. (shrink)

C W Evers, J C Walker; Knowledge, Partitioned Sets and Extensionality: a refutation of the forms of knowledge thesis, Journal of Philosophy of Education, Volume.

Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.

Current theories make a distinction between two types of case, STRUCTURAL case and INHERENT (or LEXICAL) case (Chomsky 1981), similar to the older distinction between GRAMMATICAL and SEMANTIC case (Kuryłowicz 1964).1 Structural case is assumed to be assigned at S-structure in a purely configurational way, whereas inherent case is assigned at D-structure in possible dependence on the governing predicates’s lexical properties. It is well known that not all cases fall cleanly into this typology. In particular, there is a class of (...) cases that pattern syntactically with the structural cases, but are semantically conditioned. These cases however depend on different semantic conditions than inherent cases do: instead of being sensitive to the thematic relation that the NP bears to the verbal predicate, they are sensitive to the NP’s definiteness, animacy, or quantificational properties, or to the aspectual character of the VP, or to some combination of these factors. The Finnish partitive is a particularly clear instance of this apparently hybrid category of semantically conditioned structural case. (shrink)

Partitions and their Afterlives engages with political partitions and how their aftermath affects the contemporary life of nations and their citizens.

We continue [21] and study partition numbers of partial orderings which are related to /fin. In particular, we investigate Pf, be the suborder of /fin)ω containing only filtered elements, the Mathias partial order M, and , ω the lattice of partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for Pf. We show that the partition number of is C. We also show that consistently the distributivity number of ω is (...) smaller than the distributivity number of /fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size 1 does not imply that a Polish space can be partitioned into 1 many closed sets. (shrink)

Louveau, A., S. Shelah and B. Velikovi, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 271–281. We define a notion of type of a perfect tree and show that, for any given type τ, if the set of all subtrees of a given perfect tree T which have type τ is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of type (...) τ belong to the same class. This result simultaneously generalizes the partition theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of the proof is the theorem of Halpern-Laüchli on partitions of products of perfect trees. (shrink)

The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ (...) ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (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)

This article engages with the preponderance of quotidian violence by invoking the affective registers of “trauma” and “nostalgia” in the literature on India’s Partition. It specifically attends to the category of “soft violence” and its far-reaching consequences in the Post-Partition society. The paper reads an array of short stories, beginning with “Phundane” by Sadaat Hasan Manto, intersectionally with other literary texts that deal with disfigured domestic situations as an index of the madness, raging outside. In “Phundane” the trauma (...) is etched into the bodies of​ ​women as brides, mothers, and “bearers” of aborted lives within the domestic​ ​enclosure in a world of all-engulfing terror. The other texts focus on the precipitation​ ​of slow agonising tragedy amidst intense hardship in refugee settlements which​ ​constitutes the very anathema of home. They together construct an​ affective​ arc of violence​ ​through the trappings of trauma, nostalgia, and other memories that fall between them, to show how the Partition that unmoored the intimate coordinates of relationship, is sustained in the normative experiences of the quotidian. (shrink)

This paper investigates lexical mass-to-count and count-to-mass operators in Slavic languages, primarily Russian and Ukrainian, by exploring the distribution and semantic contribution of the suffix -_in_/-_yn_. The focus is on two uses of the suffix: the singulative turns mass nouns like _gorox_ ‘pea’ into count, denoting sets of natural units (e.g., _gorošina_ ‘a pea’), and the massifier applies to count nouns, such as _kon’_ ‘horse’, and turns them into mass (e.g., _konina_ ‘horsemeat’). It is proposed that each use of -_in_/-_yn_ (...) contributes a partition operator which triggers a new division into units of the original material part. It is further argued that the singulative and the massifier should be unified, given their (i) phonological identity, (ii) shared grammatical properties, and (iii) common semantic core. Under the proposed analysis, there is a single suffix that functions as an underspecified lexical partition shifter. (shrink)

Kolodny and MacFarlane (“Ifs and oughts,” JPhil 107(3)) and MacFarlane (Assessment sensitivity, Chapter 11) describe the distinctive behavior of deontic modals in deliberation. They highlight two main features that any semantic theory should account for. First, deontic modals are informational in the sense that relativity to a body of information is an essential part of their interpretation. Second, modus ponens should be invalid, and it should fail when such modals are involved. And they argue for an account that posits information-neutral (...) semantic values. In this paper, I provide an account of deontic modals that does not involve any deviation from standard possible world semantic values, yet can capture their distinctive semantic behavior. The key element of the proposal is a partition-based semantics that treats available information as placing a constraint on interpretation. (shrink)

L'impôt sous son angle colonial en Afrique était perçu comme un instrument de pression, d'oppression et d'appauvrissement du citoyen contribuable par l'impôt dit, de capitation. Ce lourd passé historique pèse encore sur les systèmes fiscaux d'Afrique et d'ailleurs, systèmes au sein desquels les citoyens se sentent encore victimes parce que spoliés de leurs biens par les puissances publiques. Dans cet ouvrage, Docteur Célestin Amané DAGO propose un changement de paradigme du système de la fiscalité à travers la philosophie politique normative (...) et l'éthique économique appliquée du philosophe américain John Rawls, en introduisant une part de volontariat dans l'accomplissement de l'obligation fiscale, afin de changer le rapport du contribuable à l'Etat. L'ouvrage vu sous cet angle, est un appel aux pouvoirs publics pour inscrire le contribuable au coeur du système fiscal afin que, le paiement de l'impôt soit un acte spontané et volontaire. L'acte volontaire, s'entend comme la réponse à la justice et à l'équité fiscales dans la redistribution sociale des richesses nationales. Au commencement était la philosophie en tant que mère de toutes les sciences et, naturellement M. Amané Célestin DAGO, embrassa cette discipline prestigieuse à l'Université de BOUAKE ; mais, cet eudémoniste toujours souriant et généreux, est un monomane de l'application des savoirs, un citoyen rivé au credo selon lequel le bonheur de la cité ne viendra que d'un gouvernement de philosophes. Voilà le principe d'intelligibilité de toutes ses activités théoriques et pratiques. Etonné par la ruée automatique des politiciens négro-africains vers la démocratie comme régime politique idéal, parce que le maître blanc en domination l'a dit, Monsieur DAGO prépara et soutint un mémoire de maîtrise en philosophie politique et sociale sur le thème suivant : Du procès de la démocratie dans la République de PLATON ; à la suite de quoi, un mémoire de DEA de philosophie intégrale intitulé : La République comme médiation de la justice chez PLATON ; puis, voulant être différent du philosophe THALES de Milet, il prépara son second DEA en philosophie politique normative, qui a eu pour thème : La justice et la répartition fiscale dans la philosophie de JOHN RAWLS. Il est admis au Cycle Supérieur de l'Ecole Nationale d'Administration (ENA) où il obtient le Diplôme option Impôts, promotion 2004-2005. Depuis novembre 2011, Monsieur DAGO a été nommé Receveur des Impôts Divers par Arrêté du Ministre en charge de l'Economie et des Finances. Malgré ses nombreuses charges administratives, Monsieur DAGO a su réaliser sa passion en soutenant une thèse unique de Doctorat de philosophie politique normative, option éthique économique appliquée. Cette thèse a été sanctionnée par la mention très honorable à l'unanimité du jury. Ajoutons que Monsieur DAGO est l'actuel Président de la Mutuelle de Développement de HIRE (MUDH), mission de développement qui l'a contraint à écrire le livre blanc de la sous-préfecture de HIRE intitulé : Besoin de développement et Exploitation minière à HIRE. Aujourd'hui, Dr DAGO est Administrateur Principal des Sces Financiers à la Direction Générale des Impôts de Côte d'ivoire. (shrink)

This paper addresses automatic partitioning in complex reinforcement learning tasks with multiple agents, without a priori domain knowledge regarding task structures. Partitioning a state/input space into multiple regions helps to exploit the di erential characteristics of regions and di erential characteristics of agents, thus facilitating learning and reducing the complexity of agents especially when function approximators are used. We develop a method for optimizing the partitioning of the space through experience without the use of a priori domain knowledge. The method (...) is experimentally tested and compared to a number of other algorithms. As expected, we found that the multi-agent method with automatic partitioning outperformed single-agent learning. (shrink)

In the Zermelo–Fraenkel set theory (ZF), |$|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$| for any infinite set |$A$|⁠, where |$\textrm {fin}(A)$| is the set of finite subsets of |$A$|⁠, |$2^{|A|}$| is the cardinality of the power set of |$A$| and |$\textrm {Part}(A)$| is the set of partitions of |$A$|⁠. In this paper, we show in ZF that |$|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$| for any set |$A$| with |$|A|\geq 5$|⁠, where |$\textrm {Part}_{\textrm {fin}}(A)$| is the set of partitions of |$A$| whose members are finite. We (...) also show that, without the Axiom of Choice, any relationship between |$|\textrm {Part}_{\textrm {fin}}(A)|$| and |$2^{|A|}$| for an arbitrary infinite set |$A$| cannot be concluded. (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)

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ (...) is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

TheRohrlich, Fritz basic non-classical notion in quantum mechanics (QM) is the notion of superposition; entanglement is a particularly vexing special case. This chapter focuses on slowly developing the notion of a superposition state starting at the logical level of superposition applied to the notion of a set. To mathematically represent a superposition set, one must go beyond the one-dimensional notion of a 0, 1-vector representing which elements of the universe set are in the subset. A two-dimensional matrix will do the (...) job with the 0, 1-vector along the diagonal and the off-diagonal elements of 1 representing which elements are superposed in the sense of being in the same equivalence class of an equivalence relation (which is just another way to look at a partition). Those 0, 1-matrices are the incidence matrices of equivalence relations. When such logical representations of equivalence relations are normalized by dividing through by the trace (sum of the diagonal elements), one arrives at a density matrix–which is the partitional way to represent a quantum state. The three main distinctive elements of QM math are the notions of a quantum state, a quantum observable, and a (projective) measurement applying an observable to a state. This chapter gives the partition-based treatment of quantum states via density matrices. (shrink)

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered (...) partition of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

ThisGalileo new approach to understanding and interpreting quantum mechanics (QM) is based on the development of the notion of a partition on a set (or, equivalently, an equivalence relation on a set or a quotient set). The notion of a partition is not some ad hoc notion designed to make yet another interpretation of quantum mechanics. It is as fundamental a notion as that of a subset. But the math (and logic) of subsets was developed long before the (...) corresponding partition math. There is a recent sequence of developments that constitute a revival or renaissance of partitions outside of the standard treatment in enumerative combinatorics. Standard logic is the Boolean logic of subsets (propositional logic being a special case). Subsets and partitions are dual concepts in category theory. The first stage in the revival was the development of the logic of partitions dual to the logic of subsets. Then just as finite Boole-Laplace probability theory starts as the quantitative version of the logic of subsets, so the quantitative version of the logic of partitions was the second stage resulting in the new foundations for information theory based on the notion of logical entropy. The third stage was based on the realization that the distinctive mathematics of QM is the vector (particularly Hilbert) space version of the mathematics of partitions. That is the subject of this book. (shrink)

Subgraph matching on a large graph has become a popular research topic in the field of graph analysis, which has a wide range of applications including question answering and community detection. However, traditional edge-cutting strategy destroys the structure of indivisible knowledge in a large RDF graph. On the premise of load-balancing on subgraph division, a dominance-partitioned strategy is proposed to divide a large RDF graph without compromising the knowledge structure. Firstly, a dominance-connected pattern graph is extracted from a pattern graph (...) to construct a dominance-partitioned pattern hypergraph, which divides a pattern graph as multiple fish-shaped pattern subgraphs. Secondly, a dominance-driven spectrum clustering strategy is used to gather the pattern subgraphs into multiple clusters. Thirdly, the dominance-partitioned subgraph matching algorithm is designed to conduct all isomorphic subgraphs on a cluster-partitioned RDF graph. Finally, experimental evaluation verifies that our strategy has higher time-efficiency of complex queries, and it has a better scalability on multiple machines and different data scales. (shrink)

Partition regularity over algebraic structures is a topic in Ramsey theory that has been extensively researched by combinatorialists [2, 3, 5, 15]. Motivated by recent work in this area, we investigate the computability-theoretic and reverse-mathematical aspects of partition regularity over algebraic structures—an area that, to the best of our knowledge, has not been explored before. This article focuses on a 1975 theorem by Straus [25], which has played a significant role in many of the results in this field.

This article examines the problem of mental partitioning and mental conflict in Han 漢 dynasty sources. It begins by outlining two Greek psychological models—the Platonic tripartite model and the Stoic monistic model—and explains the connection between the two psychological models and their differing descriptions of mental conflict. It then analyzes passages from a seldom discussed text, the _Extended Reflections_ (_Shenjian_ 申鑒), written by the Eastern Han thinker X un Yue 荀悅. A combined analysis of the _Extended Reflections_ with fragments from (...) other Han dynasty thinkers uncovers debates on mental partitioning and mental conflict that are in some ways analogous to those found in Greek philosophical literature. One view posits that the human psyche is composed of two parts capable of producing conflicting motivations. The other view sees all actions as stemming from a single psychological center. The conclusions of this article offer new perspectives on early Chinese moral psychology that have important implications for modern research on early Chinese philosophy. (shrink)