Taking the lead from orthodox quantum theory, I will introduce a handy generalization of the Boolean approach to propositions and questions: the orthoalgebraic framework. I will demonstrate that this formalism relates to a formal theory of questions (or ‘observables’ in the physicist’s jargon). This theory allows formulating attitude questions, which normally are non-commuting, i.e., the ordering of the questions affects the answer behavior of attitude questions. Further, it allows the expression of conditional questions such as “If Mary reads the book, (...) will she recommend it to Peter?”, and thus gives the framework the semantic power of raising issues and being informative at the same time. In the case of commuting observables, there are close similarities between the orthoalgebraic approach to questions and the Jäger/Hulstijn approach to question semantics. However, there are also differences between the two approaches even in case of commuting observables. The main difference is that the Jäger/Hulstijn approach relates to a partition theory of questions whereas the orthoalgebraic approach relates to a ‘decorated’ partition theory (i.e. the elements of the partition are decorated by certain semantic values). Surprisingly, the orthoalgebraic approach is able to overcome most of the difficulties of the Jäger/Hulstijn approach. Furthermore, the general approach is suitable to describe the different types of (non-commutative) attitude questions as investigated in modern survey research. Concluding, I will suggest that an active dialogue between the traditional model-theoretic approaches to semantics and the orthoalgebraic paradigm is mandatory. (shrink)

Introduction -- The fundamentals and their role in design -- Underneath it all -- Tiling the plane without gap or overlap -- Symmetry, patterns and fractals -- The stepping stone of Fibonacci and the harmony of a line divided -- Polyhedra, spheres and domes -- Structures and form in three dimensions -- Variations on a theme: modularity, closest packing and partitioning -- Structural analysis in the decorative arts, design and architecture -- A designer's framework.

Located in Kleinbasel close to the Rhine, the Kaskadenkondensator is a place of mediation and experimental, research-and process-based art production with a focus on performance and performative expression. The gallery, founded in 1994, and located on the third floor of the former Sudhaus Warteck Brewery (hence cascade condenser), seeks to develop interactions between artists, theorists and audiences. Eight, maybe, nine or ten 40 litre bags of potting compost lie strewn about the floor of a high-ceilinged white washed hall. Dumped, split (...) open, the soil mixed with iridescent specks of green, blue and red glitter. On the walls hang large black and white photographic images—negative and positive prints barely clean, hardly sharp, scavenged from the world and presented half processed. On a third wall, hangs a framed golden and charcoal surface. Finally, a huge stain of black dye runs down a wall that descends into a sunken quarter of the Kaskadenkondensator gallery space. The results of a collaboration between Oliver Minder and Walter Derungs reflect on themes addressed in the recent Aesthetics in the 21st Century conference held by the department of English, University of Basel. In particular, the joint show questions how an aesthetic experience may be other than a human-world interaction, hinting at the withdrawal and veiling that objects perform, while demanding that different works engage with each other and play out this game under the non-supervisory eyes of a human audience. Things here are becoming—sometimes it’s a movement towards a more complete ontic whole in a projection of finality, other times it’s a dispersal, an atrophy to rather disarrayed entities. Yet, in the moment and place in which the objects are, we take them as here and now. Let’s get to the material of the stuff that Minder and Derungs have assembled. Oliver Minder employs organic materials—potting earth, cuttlefish ink secretion, rice, and insects; yet his works hardly seem natural in the sense of a harmonic relation between material and the form they are constrained into, the objects they are compelled to occupy. For the substrates on, through, or within which these natural materials are mediated are harshly inorganic substances—Plexiglas, safety glass, acrylic resin, boat varnish, spray paint. Minder, thus, generates a conflict within the materiality of his work between two polar opposites—from the human perspective—in the contiguity of materials engaging with each other in a thrown together formation that nonetheless appears to keep the materials and the objects they make in happy accidental relation to each other. Let me expand a little: on the one hand, the things Minder makes query our belief in substance as belonging in a particular domain, an environment suited to precisely that stuff. We are focused on thinking categorically where things belong, both in terms of natural place and natural relations they might extend to each other. Hence, we are driven to think of environment and order. On the other hand, while extracting things from their conventional place and arranging them within awkward constellations that we as observers feel isn’t quite right, Minder manages to persuade the viewer that the materials are nonetheless “doing alright.” So, simultaneous to our awareness of the appropriateness of the world according to our global notions of accord and uniformity, we are forced to accept the local discrepancies of disassociation, inappropriateness and misplacement. The tension between these two vectors generates a vacillation that intensifies Minder’s work. In the Kaskadenkondensator works, then, it is vital to first consider the material of Minder’s works: potting compost—what is it doing here in the first place?—seems to enjoy being “polluted” by sparkly glitter. Glitter has a long history, used in cosmetics by the Egyptians, and in cave paintings, too, earlier made of beetle shells and mica, nowadays glitter is made of plastic cut to minute sizes down to 50 microns. So what’s the point here? Well shiny bits of dust-like material are actually generated from ultra-thin plastic sheets and are normally cut into shapes that fit contiguously on a two-dimensional surface: squares, triangles, hexagons etc. What then appears to be totally random, chaotic decoration, is actually an array of extremely regular identifiable objects. 1 Of course scale has a role to play here. The minuteness of the dimensions means the regularity is beyond our recognition—all we see are the twinkling surfaces of the multi-coloured grains of plastic. In contrast, potting compost, which appears to be unary in its dull unresponsive lumpen disposition, is in fact an amalgam of a variety of organic and inorganic materials: peat, bark, mushroom compost, and sand and perlite, and should perhaps be more proactively exciting to the viewer because of this complexity. Yes; we can (if we care to) identify different textures, different sizes in the mixture of the medium, but I claim that we tend to treat this organic/inorganic assemblage as just a simple substance. Further and crucially important to our consideration here is that the medium is partially contained, but also partially spilling from the split plastic bags in which it is sold in garden centres. That the compost spills out gives it a movement suggesting life; that the bags are cast here and there in a random fashion by Oliver Minder, lying like discarded carcasses, hacked torsos, dismembered bodies, suggests a horrific murder scene, a Tatort. 2 The glitter flourishes in the medium, lies happy and decorative; that is simply what it does, how it is—always already broken, made-for-scattering, designed to be incomplete; the taken-to-be-natural compost, in contrast, cannot rest content but is forced to speak to us metaphorically in its abject overflowing of violence and rupture. While Oliver Minder’s elements in the installation direct our attention to material, Walter Derungs’ works raise questions around seeing and making in photography. There is a simultaneous flicker between the materials and their use in the production of a sense making representation, on the one hand, and on the other the very notion of what is worthy of picturing, framing, representing on the other. Derungs' images are of non-places. Ranging from archaic decaying monster buildings, buildings that have gone far beyond the ravages of a time that we can safely associate with the genteel preservation of a Bernd and Hill Belcher post-industrial decline, to the background “noise” of an urban world that is falling apart, and to which we most of the time seem to pay little attention, and habitually just pass by. In this respect, their non-ness differs somewhat from the conventional association of the term with Marc Augé 3 , where emphasis is on the specifics (if we do care to examine them for their non-placedness) of the spatial or place containment in which movement between multimodal coordinates occurs in supermodern late capitalist post-urban spaces. In other words, we might be in an Augéian non-place and (not) experience—be impervious to—that environment, or we might in Derungs’ manner look out from such a position at the “scenery” around us. I claim scenery, as this is what Derungs seems to do with his partial photography—construct a very purposefully articulated, symmetric, flat world of image. Mostly depopulated, his images construct a space in which the direction of time is uncertain: are these partial structures falling apart, or perhaps terminated in a never-to-be-completed state, or are they a few steps from final completion? Temporal and spatial dimensions figure large in Derungs’ image-making: his world, and perhaps this is in fact the only way for it to be registered photographically, is already image before it is photographed. A key combination of images in this show is a matrix of six black and white negative prints measuring 300 x 215 cm that form the image of a semi-derelict (or is it yet incomplete) church, and adjacent on a perpendicular wall, a single black and white positive print 150 x 250 cm of two bricked-up windows of a late-Victorian industrial building. What are we led to believe that we see here? In the negative print, the conditions of perception 4 are sufficiently reproduced for us to recognise the structure of the building, to distinguish ground and form, to relate some partial elements of narrative, and to recognize symbols such as the alter cross and figure of Christ, a looming crane, a traffic cone, and banks of tiered seating. We piece the image together both from the individual forms which we recognize despite the tonal reversal, and we piece the six prints together as a whole, the matrix of lines between them emphasizing our purview onto the world. While we recognise the forms at work in the image and might possibly relate the negative reversals to other figurations such as Vera Lutter’s camera obscura exposures, we cannot but avoid seeing the partialness of the image in the sponge marks of the developer that was spread by hand across the prints. 5 Derungs’ thus intervenes with our usual conception of photography as the mimetic realist vehicle sine qua non , by exposing the viewer to tonal reversal and incomplete or over developed areas of the print. We thus confront both the idiom of such image making and its raw (chemical) materiality at once in the simultaneity of the recognition of what the image pictures and the recognition that it is in the act of picturing. The church image, taken from the series “BW Negativs 2011,” thus orients us towards how we see things in the world via photographs. The single image of the bricked-up wall presents us with a completely different visuality that relates to a faciality 6 which we cannot easily escape from. We look, or rather try to look with no success, through the face of the windows, through the classic Albertian screen 7 which has already been given to us in the church matrix beside. Yet although we should be able to make more of these concealed windows because they are a positive print, because they are complete, because they approach us on a more realistic scale, reproduced at life size, we cannot. The objects pictured here withdraw from us; furthermore, they merely mock our blindness at not seeing how we look. Blocked up with quite a hint of paned glass behind, one window is blanked out with a white blind, the other simply blankly dark. The apertures look like eyes with teeth in them, or a Dogon mask, or even Man Ray’s Noire et Blanche (1926) if we want to get really perverse. The height of elegant modernist chauvinist beauty thrown against the vacuity of post-industrial decline. Derungs thus catapults us consciously into a world enfolded with and through images, but in such a way that the images themselves become objects that stand resistant to us, impervious to our gaze, indifferent. We—and indeed they—do not attempt to reach out to a real that is beyond, rather the images play in a world that is just theirs, and we can only enter that world if we too submit to their regime: tonal reversal, segmented, partial, inadequate, still, wrenched out of time. In contemplation, in the flood of the image “falling” off the wall, we too become image-object. Perhaps enough has now been said about the works, yet enough can never really be said, we know the image will always exceed the word—let’s accelerate the critique: Derungs’ work continues in a second space partially partitioned from this first room. Opposing three more “BW Negativs” which figure yet more quotidian aspects of the world is Minder’s gold spray paint and cuttlefish secretion mix: things that just shouldn’t work together do in the dialogue between stuff that Derungs and Minder have constructed. Minder makes things; Derungs makes images; together they make objects which inhabit their own world which we can approach and sensually engage with and come to grips with only on those objects’ own terms. This is best summarised by a final work made by Oliver Minder which on a third wall faces these two semi-partitioned spaces. A deep black stain about 100 X 200 cm with streak marks running down a further 2 metres hovers positioned to observe the whole work, and also to be part of this installation, too. This liminal flat suzerain lies in/out of the whole work. The stain of cuttlefish secretion resonates with Derungs’ sponge strokes on the church image; it mirrors the iris of an all-seeing eye; it combines material in situ with the situation itself. Where Minder’s other works have material and medium or substrate upon which the material is exercised, this single black hole is image which sucks everything up into itself. It draws the viewer, who must otherwise look away attentively at the floor work, and imagine horror, or smile at the ironic play of glitter. Look away at the image constructions that suggest how it is we too look to our world. See the play of thing and image in a third area. Or, finally return to the base of the pyramid that triangulates, to realise the stuff-image that unlocks it all for us. Black on white, organic on inorganic, material to substrate, that which in the falling out of one on the other, in its running down the wall simply gives form to both content and expression in one direction, and content and expression to form in another. NOTES In fact, glitter is used as associative forensic evidence: the 20,000 or so varieties are all uniquely identifiable. Joel Sternfeld, Tatorte: Bilder gegen das Vergessen (München: Schirmer/Mosel, 1996). Marc Augé, Non­places: Introduction to an Anthropology of Supermodernity trans. John Howe (London: Verso, 2006). An echo of the uneven paint strokes of light­sensitive chemicals in the paper preparations made by Henry Talbot some 170 years ago in the first sun drawings that also often pictured architectural forms. It was Talbot’s surprising discovery that where a weaker chemical solution was more thinly spread, greater light sensitivity was actualized, yet this virtual image had then to be chemically developed in a second step. Thus, Derungs unevenly finished spongings suggestively trace back to this originary technology (although his sweeps are the stains of uneven development and not those of the initial preparation of light–sensitive material). Umberto Eco, “Critique of the Image” in “Articulations of Cinematic Code” Cinematics 1, 1970. Gilles Deleuze and Felix Guattari, A Thousand Plateaus: Capitalism and Schizophrenia trans. Brian Massumi (Minneapolis: University of Minnesota Press, 2007). Geoffrey Batchen, Burning With Desire: the Conception of Photography (Cambridge, Mass: MIT Press, 1999). (shrink)

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.

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

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 $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$.

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)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

In this paper we examine partitioned interpretations of sentences with reciprocal expressions. We study the availability of partitioned readings with definite subjects and proper name conjunctions, and show new evidence that partitioned interpretations of simple reciprocal sentences are independent of the semantics of the reciprocal expression, and are exclusively determined by the interpretation of the subject.

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)

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)

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.

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)

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)

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.

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)

The last section made it clear that an analysis which at first seems to fail is viable after all. It is viable if we let it depend on a partition function to be provided by the context of conversation. This analysis leaves certain traits of the partition function open. I have tried to show that this should be so. Specifying these traits as Pollock does leads to wrong predictions. And leaving them open endows counterfactuals with just the right amount of (...) variability and vagueness. (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)

Given that the world as we perceive it appears to be predominantly classical, how can we stabilize quantum effects? Given the fundamental description of our world is quantum mechanical, how do classical phenomena emerge? Answers can be found from the analysis of the scaling properties of modular quantum systems with respect to a given level of description. It is argued that, depending on design, such partitioned quantum systems may support various functions. Despite their local appearance these functions are emergent properties (...) of the system as a whole. With respect to the separation of subject and object such functions of interest are control, simulation, and observation. They are interpreted in close analogy with more basic physical behavior. (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.

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)

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 Decoration on the Cult Chapel Walls of the Old Kingdom Tombs at Giza: A New Approach to Their Interaction. By Leo Roeten. Culture & History of the Ancient Near East, vol. 70. Leiden: Brill, 2014. Pp. lix + 436, illus. $218.

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)

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.

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)

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

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.

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 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)

The goal of this short note is to interest set theorists in the order type ω*ω1, and to encourage them to work on the question of whether or not the Continuum Hypothesis decides the partition relation τ→2, for τ=ω*ω1 and for τ=ω1ω+2.

We will present the main characteristics of the decorative scheme from throne room in the palace of the Neo-Assyrian king Ashurnasirpal II, whose reign extended from 883-859 BC, located in the ancient city of Kalhu, now north Iraq. We consider decorative scheme to be the presence of images and texts in an architectural setting. We believe that the creation of the decorative scheme may be in some way linked to political projects, and therefore, it would be an expression of the (...) royal ideology and rhetoric. Throughout the analysis of the decorative scheme, aided by textual and material sources, we will try to put into evidence the main aspects of the imperial policy in vigour. (shrink)

A desirable property of a diversity index is strict concavity. This implies that the pooled diversity of a given community sample is greater than or equal to but not less than the weighted mean of the diversity values of the constituting plots. For a strict concave diversity index, such as species richness S, Shannon''s entropy H or Simpson''s index 1-D, the pooled diversity of a given community sample can be partitioned into two non-negative, additive components: average within-plot diversity and between-plot (...) diversity. As a result, species diversity can be summarized at various scales measuring all diversity components in the same units. Conversely, violation of strict concavity would imply the non-interpretable result of a negative diversity among community plots. In this paper, I apply this additive partition model generally adopted for traditional diversity measures to Aczél and Daróczy''s generalized entropy of type . In this way, a parametric measure of -diversity is derived as the ratio between the pooled sample diversity and the average within-plot diversity that represents the parametric analogue of Whittaker''s -diversity for data on species relative abundances. (shrink)

The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.

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)

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)

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).

For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction of an (...) interval algebra in which every well-ordered chain has size less than cmm. (shrink)

It is known that in an infinite very weakly cancellative semigroup with size $\kappa $, any central set can be partitioned into $\kappa $ central sets. Furthermore, if $\kappa $ contains $\lambda $ almost disjoint sets, then any central set contains $\lambda $ 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 ^\omega = \kappa $, then the statement holds for J-sets. (shrink)

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

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal family (...) of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (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)

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)

Fragments of painted plaster found in the destruction layer of three first floor rooms in the House of Seals and of one first floor room of the House of the Sword had broken from the ceiling. They allow the restoration of the schema in the room of the House of the Sword and of two of the rooms in the House of Seals, the latter in colour. Composed of bands surrounding a quadrangular field, these décorations, evoking carpets stretched on the (...) ceiling, are richer (costly pigments, gold leaf) and more complex (some fifteen motifs) than those in tombs of the same period, which adopted the same « carpet » composition and which borrowed from the same repertoire, often confirmed by the textiles. Since we know of no other ceiling decoration in habitation rooms, it is unwise to contrast the exuberance of these ceilings with the tomb ceilings. The room that could not be restored had moreover a much more perfunctory decoration. (shrink)