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)

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)

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)

A new logic of partitions has been developed that is dual to ordinary logic when the latter is interpreted as the logic of subsets of a fixed universe rather than the logic of propositions. For a finite universe, the logic of subsets gave rise to finite probability theory by assigning to each subset its relative size as a probability. The analogous construction for the dual logic of partitions gives rise to a notion of logical entropy that is precisely related (...) to Claude Shannon's entropy. In this manner, the new logic of partitions provides a logico-conceptual foundation for information-theoretic entropy or information content. (shrink)

We study tree-like decompositions of models of a theory and a related complexity measure called partition width. We prove a dichotomy concerning partition width and definable pairing functions: either the partition width of models is bounded, or the theory admits definable pairing functions. Our proof rests on structure results concerning indiscernible sequences and finitely satisfiable types for theories without definable pairing functions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

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)

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)

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)

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)

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)

We define two versions of stability and efficiency of partitions and analyze their relationships for some matching rules. The stability and efficiency of a partition depends on the matching rule φ. The results are stated under various membership property rights axioms. It is shown that in a world where agents can freely exit from and enter coalitions, whenever the matching rule is individually rational and Pareto optimal, the set of φ-stable and φ-efficient partitions coincide and it is unique: the (...) grand coalition. Then we define a weaker version of stability and efficiency, namely specific to a given preference profile and find some negative results for stable matching rules. (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)

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)

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)

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of (...) questions for the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be ${\Pi^{0}_{1}}$ -, ${\Pi^{0}_{2}}$ -, and ${\Sigma^{1}_{1}}$ -complete, respectively. (shrink)

Generalizing model companions from model theory we define companions of pieces of canonical partitions of Polish G-spaces. This unifies several constructions from logic. The central problem of the paper is the existence of companions which form a G-orbit which is a Gδ-set. We describe companions of some typical G-spaces.

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 advocate of modal logic or relevant logic has traditionally argued that her preferred system offers the best regimentation of the theory of entailment. Essential to the projects of modal and relevant logic is the importation of non-truth-functional expressive resources into the object language on which the logic is defined. The most elegant technique for giving the semantics of such languages is that of frame semantics, a variation on which features the device of partitioned frames that divide 'points of (...) evaluation' into two types: normal points and abnormal points. This essay attempts to provide satisfying philosophical interpretations of the partitioned frame semantics for some weak modal logics and relevant logics. According to the current best interpretation, partitioned frame semantics are a kind of possible worlds semantics that appeal to 'logically impossible worlds' where the laws of logic may fail to hold. This is undermined by the fact that all of the target logics fail to satisfy criteria of expressibility needed to support the thesis that laws of logic fail to hold at abnormal points. This suggests that we should not look for a monistic interpretive paradigm for partitioned frame semantics. (shrink)

Our object is to study the interaction between mereology and David Lewis’ theory of subject-matters, elaborating his observation that not every subject matter is of the form: how things stand with such-and-such a part of the world. After an informal introduction to this point in Section 1, we turn to a formal treatment of the partial orderings arising in the two areas – the part-whole relation, on the one hand, and the relation of refinement amongst partitions of the set (...) of worlds, on the other. We emphasize a certain duality, formulated in and in Section 2, between the corresponding lattice operations – mereological joins with partition-lattice meets, mereological meets with partition-lattice joins. Section 3 presents some issues that are raised by consideration of the informally familiar idea of logical subtraction. These include, in particular, a problem about the need for a notion of independence different from the usual logical notion going by that name. The apparatus of Section 2 promises to throw some light on this problem, as we indicate in Section 4. Section 5 ties up some loose ends and suggests an area in which further work would be desirable. (shrink)

A strong coloring on a cardinal$\kappa $is a function$f:[\kappa ]^2\to \kappa $such that for every$A\subseteq \kappa $of full size$\kappa $, every color$\unicode{x3b3} <\kappa $is attained by$f\restriction [A]^2$. The symbol$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$asserts the existence of a strong coloring on$\kappa $.We introduce the symbol$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$which asserts the existence of a coloring$f:[\kappa ]^2\to \kappa $which isstrong over a partition$p:[\kappa ]^2\to \theta $. A coloringfis strong overpif for every$A\in [\kappa ]^{\kappa }$there is$i<\theta $so that for every color$\unicode{x3b3} <\kappa (...) $is attained by$f\restriction ([A]^2\cap p^{-1}(i))$.We prove that whenever$\kappa \nrightarrow [\kappa ]^2_{\kappa }$holds, also$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$holds for an arbitrary finite partitionp. Similarly, arbitrary finitep-s can be added to stronger symbols which hold in any model of ZFC. If$\kappa ^{\theta }=\kappa $, then$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$and stronger symbols, like$\operatorname {Pr}_1(\kappa,\kappa,\kappa,\chi )_p$or$\operatorname {Pr}_0(\kappa,\kappa,\kappa,\aleph _0)_p$, also hold for an arbitrary partitionpto$\theta $parts.The symbols$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$hold for an arbitrary countable partitionpunder the Continuum Hypothesis and are independent over ZFC$+ \neg $CH. (shrink)

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B |<κ(T ) pieces, Ai | i < |B |<κ(T ) , such that for each Ai there is a Bi ⊆ B where |Bi| < κ(T ) and Ai..

Working in ZFC, we show that for any infinite cardinal κ and ordinal $\gamma the polarized partition relation $\[\begin{pmatrix} (2^{ → $\[\begin{pmatrix}(2^{ holds. Our proof of this relation involves the use of elementary substructures of set models of large fragments of ZFC.

Özkal-Sanver (Theory Decis 59:193–205, 2005) studies stability and efficiency of partitions of agents in two-sided matching markets in which agents can form partitions by individual moves only, and a matching rule determines the matching in each coalition in a partition. In this study, we present the relationship between stability and efficiency of partitions that is analyzed for several matching rules and under various membership property rights codes, now allowing coalitional moves.

This article proposes a model for route description dialogues based on the integration of the theories about route directions with those related to specific spatial instructions. This proposal is based empirically on a corpus of spoken Italian. The analysis of the corpus has shown that the basic dialogue strategy proposed by previous researchers requires further integration and elaboration. Also the model of spatial language developed in cognitive linguistics applies to the description of the giver's instructions and directions, despite the fact (...) that the literature in the field often overlooks this approach. This way of accounting for route descriptions is suitable to explain the lexicalization of the conceptualization of the displacements the follower must accomplish. A model of dialogue partitions integrated with the cognitive modelling of spatial language is proposed and supported by examples from the linguistic data. (shrink)

It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal of nonstationary subsets (...) ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+ −I), then anI-partitionofAis a maximal collectionW⊆,P ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y.. (shrink)

Does the soul have parts? What kind of parts? And how do all the parts make together a whole? Many ancient, medieval and early modern philosophers discussed these questions, thus providing a mereological analysis of the soul. The eleven chapters reconstruct and critically examine radically different theories. They make clear that the question of how a single soul can have an internal complexity was a crucial issue for many classical thinkers.

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)

Working in the theory “ZF + There is a nontrivial elementary embedding j: V → V ”, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every (...) cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation μ → [μ]ωℵ2. We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j : V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice. (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)

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice , we study partitions of Russell-sets into sets each with exactly n elements , for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is (...) not divisible by any finite cardinal n > 1. (shrink)

We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.

This article aims to rethink the trauma of the 1947 Partition of British India through the figure of the border. It is at the border that we can see how the present is as much constituted by the concentration of new realities that call for shifting frameworks of understanding as it is by past events that continue to haunt memory. It undertakes this task through a close reading of the trope of borders in Saadat Hasan Manto’s 1953 short story, (...) ‘Toba Tek Singh’. Partition fiction serves as a fruitful ground for developing new approaches to that history because, in part, literature is uniquely situated between representation and theory, between what a text represents and how it represents. In Manto’s story, the border is embodied in its central protagonist, Bishan Singh, who experiences an ontological struggle between being and belonging that exceeds the particular historical, geographical, and national context of the Partition. I suggest that the division of British India signaled a unique rupture within the subcontinent in which the creation of borders became the defining traumatic event of that history. This article thus moves away from the familiar nationalist rhetoric that has otherwise dominated conversations about the 1947 Partition of British India. It focuses instead on the geographical border as a conceptual figure that is at once spatially and corporeally oriented to a collective identity that exceeds nationalist frameworks of reference. (shrink)

Scientific theories of consciousness identify its contents with the spatiotemporal structure of neural population activity. We follow up on this approach by stating and motivating Dynamical Emergence Theory, which defines the amount and structure of experience in terms of the intrinsic topology and geometry of a physical system’s collective dynamics. Specifically, we posit that distinct perceptual states correspond to coarse-grained macrostates reflecting an optimal partitioning of the system’s state space—a notion that aligns with several ideas and results from computational (...) neuroscience and cognitive psychology. We relate DET to existing work, offer predictions for empirical studies, and outline future research directions. (shrink)

This article studies the narratives of voices identifying the harrowing aftermath of the 1947 partition of India and Pakistan, and the representations of the contemporary effects of partition in Bapsi Sidhwa’s novel Ice Candy Man. The narrative unfolds past experiences through the eyes of different characters and surroundings from different social, political and religious backgrounds. The novel vividly portrays the horror of violence during the partition, as communities that once coexisted peacefully become engulfed in a whirlwind of (...) hatred and bloodshed. Sidhwa’s polyphonic narrative experiences reflect the multifaceted nature of violence, from communal riots to personal tragedies creating a poignant portrayal of human suffering and resilience. The theme of partition, violence, and memory permeates throughout the novel, encapsulating the socio-political upheavals that marked the birth of two nations – India and Pakistan. This paper, therefore, attempts to analyse the novel in order to understand the ideas of the ‘polyphonic narrative’ of Mikhael Bakhtin’s theory and many other postcolonial critic’s ideas. For portraying the human experience amidst a backdrop of partition and historical turmoil in shaping narratives as presented in the text by applying the theoretical framework of Postcolonial theory. (shrink)

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

We present a well-defined generalisation of the core to coalitional games with externalities, where the value of a deviation is given by an endogenous response, the solution (if nonempty: the core) of the residual game.

Assuming $V=L+AD$, using methods from inner model theory, we give a new proof of the strong partition property for ${\sim}{ \delta }^{2}_{1}$. The result was originally proved by Kechris et al.

This paper offers a new unified theory about the meaning of the imperfective and progressive aspects that builds on earlier of analyses in the literature that treat the imperfective as denoting a universal quantifier (e.g. Bonomi, Linguist Philos, 20(5):469–514, 1997; Cipria and Roberts, Nat Lang Semant 8(4):297–347, 2000). It is shown that the problems associated with such an analysis can be overcome if the domain of the universal quantifier is taken to be a partition of a future extending (...) interval into equimeasured cells. Treating the partition-measure (the length of each partition-cell) as a contextually dependent variable allows for a unified treatment of the habitual and event-in-progress readings of the imperfective. It is argued that the contrast between the imperfective and the progressive has to do with whether the quantifier domain is a regular partition of the reference interval or a superinterval of the reference interval. (shrink)

Contemporary academic moral theory is a territory partitioned between a number of highly professionalised and (on the face of it) fiercely opposed schools of thought—consequentialism, Kantianism, virtue ethics, contractualism, natural law theory, sentimentalism and others. Not every academic ethicist is aligned with any of these schools, but most are, and all face insistent pressure to become aligned. (For example, appointing committees for ethics jobs often ask “What sort of ethicist are you?”, and tend, both intentionally and unintentionally, to (...) penalise complex or unusual answers.). (shrink)

The Knowability Paradox is a logical argument that, starting from the plainly innocent assumption that every true proposition is knowable, reaches the strong conclusion that every true proposition is known; i.e. if there are unknown truths, there are unknowable truths. The paradox has been considered a problem for every theory assuming the Knowability Principle, according to which all truths are knowable and, in particular, for semantic anti-realist theories. A well known criticism to the Knowability Paradox is the so called (...) restriction strategy. It bounds the scope of the universal quantification in (KP) to a set of formulas whose logical form avoids the paradoxical conclusion. Specifically, Tennant suggests to restrict the quantifier in (KP) to propositions whose knowledge is provably inconsistent. He calls them Anti-Cartesian propositions and distinguished them in three kinds. In this paper we will not be concerned with the soundness of the restriction proposal and the criticisms it has received. Rather, we are interested in analyzing the proposed distinction. We argue that Tennant’s distinction is problematic because it is not completely clear, it is not grounded on an adequate logical analysis, and it is incomplete. We suggest an alternative distinction, and we give some reasons for accepting it: it results logically grounded and more complete than Tennant’s one, inclusive of it and independent from non-epistemic notions. (shrink)