Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. If [math] is a closed Noetherian graph on a [math]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[math][math][math]DC that [math] is countably chromatic and there is no Vitali set.

It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $\mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $\mathbb {R}^3$ has uncountable chromatic number.

If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.

We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem involving the notion of coloring number. Classically, the coloring number of a graph $G=$ is the least cardinal $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ many vertices connected to it by $E$. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union of $n$ (...) forests, then the coloring number of the graph is at most $2n$. We focus on the case when $n=1$. (shrink)

The idea that an utterance of a basic (nondeviant) declarative sentence expresses a single true-or-false proposition has dominated philosophical discussions of meaning in this century. Refinements aside, this idea is less of a substantive theses than it is a background assumption against which particular theories of meaning are evaluated. But there are phenomena (noted by Frege, Strawson, and Grice) that threaten at least the completeness of classical theories of meaning, which associate with an utterance of a simple sentence a truth-condition, (...) a Russellian proposition, or a Fregean thought. And it may well be the case that a framework within which utterances express sequences of propositions provides much of what is needed to account for the relevant phenomena, a better overall picture of the way language works, and an enticingly uniform perspective on a variety of semantic problems. I do not myself take to theories that multiply propositions by appealing to propositions “presupposed” or to pairs of Fregean and Russellian propositions, or theories that show no respect for a distinction between semantics and pragmatics— where the former is the study of propositions whose general form and character is determined by word meaning and syntax—or for theories that blithely abandon general principles of composition and semantic innocence. I would like to sketch a package based on four interconnected ideas: (i) the meaning of an individual word is a sequence of instructions for generating a sequence of propositions (in conjunction with compositional instructions (syntax) and elements of context); (ii) utterances themselves are not bearers of truth or falsity; (iii) judgements of truth, falsity, commitment, and conflict are shaped, in part, by the weights attached to individual 1 propositions that occur in sequences expressed by utterances, weights that may be set (and reset) by contextual considerations; (iv) Fregean senses are superfluous; propositions might as well be Russellian (Mont Blanc and all its snow fields will do as well as any mode of presentation).. (shrink)

We prove a partition theorem for analytic sets, namely, if X is an analytic set in a Polish space and [X]n = K0 ∪ K1 with K0 open in the relative topology, and the partition satisfies a finitary condition, then either there is a perfect K0-homogeneous subset or X is a countable union of K1-homogeneous subsets. We also prove a partition theorem for analytic sets in the three-dimensional case. Finally, we give some applications of the theorems.

Improving a theorem of Gasarch and Hirst, we prove that if 2 ≤ k ≤ m < ω, then the following is equivalent to WKL0 over RCA0 Every locally k-colorable graph is m-colorable.

A function is diagonally non-computable if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function implies Ramsey-type weak König’s lemma. In this paper, we prove that for every computable order h, there exists an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -model of h-DNR which is not a not (...) model of the Ramsey-type graph coloring principle for two colors and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -models. (shrink)

Let ⪯R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order . We show that ⪯R has fairly high complexity with respect to Borel reducibility , although its exact classification remains open.

Many scholars claim that Frege's theory of colouring is committed to a radical form of subjectivism or emotivism. Some other scholars claim that Frege's concept of colouring is a precursor to Grice's notion of conventional implicature. I argue that both of these claims are mistaken. Finally, I propose a taxonomy of Fregean colourings: for Frege, there are purely aesthetic colourings, communicative colourings or hints, non-communicative colourings.

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of 1-dense real order types, Ann. Pure Appl. Logic 29 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in (...) Cichoń’s diagram are at least 2. (shrink)

Autism spectrum disorders reduce one’s ability to act appropriately in social situations. Increasing evidence indicates that children with ASD might ignore nonverbal social cues that usually aid social interaction because they do not recognize or understand them. We asked children with ASD to color an augmented reality coloring book to teach them how to recognize and understand some specific social signals and to ignore others. ARCB materials teach children to recognize and understand social signals in various ways. They can, for (...) example, view 3D animations of the ARCB materials on a tablet computer. Thus, the ARCB can be used to help children with ASD focus their attention on the meaning and social value of nonverbal behaviors in specific social situations. The ARCB has multiple functions: it extends the social features of the story, and it restricts attention to the most important parts of the videos. Single-subject research with a multiple-baselines across-subject design was used in this study. After five weeks of ARCB training intervention, all 3 participants’ scores rose significantly and dramatically during the intervention phase, and remained significantly higher in the maintenance phase than at baseline. We conclude that coloring pictures of social situations may help children with ASD recognize and better understand these situations. (shrink)

Recent evidence in cognitive neuroscience indicates that the visual system is influenced by the outcome of an early appraisal mechanism that automatically evaluates what is seen as being harmful or beneficial for the organism. This indicates that there could be valence in perception. But what could it mean for one to see something positively or negatively? Although most theories of emotions accept that valence involves being related to values, the nature of this relation remains highly debated. Some explain valence in (...) terms of evaluative content, others in terms of evaluative attitude. Here I shall argue that an account of affective perception in terms of attitude has more chance of succeeding. To do so, I will first highlight the difficulties that a content-based approach faces, considering the many forms that it might take. I will conclude that seeing the world positively or negatively involves more than a positive or negative content; it involves a distinctive attitude, but which one? Should it be conceived of in imperative or evaluative terms? And what makes this attitude distinct from a proper emotion? (shrink)

We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and , then there is a regular cardinal θ<μ such that any fewer than cf stationary subsets of must reflect simultaneously.

In this work, we focus on the problem of selecting low-level heuristics in a hyperheuristic approach with offline learning, for the solution of instances of different problem domains. The objective is to improve the performance of the offline hyperheuristic approach, identifying equivalence classes in a set of instances of different problems and selecting the best performing heuristics in each of them. A methodology is proposed as the first step of a set of instances of all problems, and the generic characteristics (...) of each instance and the performance of the heuristics in each one of them are considered to define the vectors of characteristics and make a grouping of classes. Metalearning with statistical tests is used to select the heuristics for each class. Finally, we used the Naive Bayes to test the set instances with k-fold cross-validation, and we compared all results statistically with the best-known values. In this research, the methodology was tested by applying it to the problems of capacitated vehicle routing and graph coloring. The experimental results show that the proposed methodology can improve the performance of the offline hyperheuristic approach, correctly identifying the classes of instances and applying the appropriate heuristics in each case. This is based on the statistical comparison of the results obtained with those of the state of the art of each instance. (shrink)

Studies of linguistic synaesthesias in English have shown a range of fine-grained language mechanisms governing the associations between colours on the one hand, and graphemes, phonemes and words on the other. However, virtually nothing is known about how synaesthetic colouring might operate in non-alphabetic systems. The current study shows how synaesthetic speakers of Mandarin Chinese come to colour the logographic units of their language. Both native and non-native Chinese speakers experienced synaesthetic colours for characters, and for words spelled in (...) the Chinese spelling systems of Pinyin and Bopomo. We assessed the influences of lexical tone and Pinyin/Bopomo spelling and showed that synaesthetic colours are assigned to Chinese words in a non-random fashion. Our data show that Chinese-speaking synaesthetes with very different native languages can exhibit both differences and similarities in the ways in which they come to colour their Chinese words. (shrink)

A structure A = (A; E₀, E₁ , . . . , ${E_{n - 2}}$) is an n-grid if each E i is an equivalence relation on A and whenver X and Y are equivalence classes of, repectively, distinct E i and E j , then X ∩ Y is finite. A coloring χ : A → n is acceptable if whenver X is an equivalence class of E i , then {ϰ Є X: χ(ϰ) = i} is finite. If (...) B is any set, then the n-cube B n = (B n ; E₀ E₁ , . . . , ${E_{n - 2}}$) is considinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpiński [17], proved that ℝ n has an acceptable coloring iff ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$. The main result is: of A is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if A embeds all finite n-cubes, then ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$; if A embeds ℝ n , then ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$; (3) A has an acceptable coloring. (shrink)

The normativity objection challenges normative naturalism by arguing that we have a distinctive cognitive experience when making normative judgements, finding ourselves in touch with some action-guiding authority issuing demands from outside, and that this cannot be explained naturalistically. An increasing number of naturalists have defended their position by adopting the coloring strategy, which aims to explain away the need for positing a special property and contends that the normative feel results from the intricate work of our mind which colors the (...) world. In this paper, I critically review the extant strategies and consider what the most plausible form of the strategy would look like. I further argue that even the strategy in its most plausible form faces serious problems, that it lacks positive motivation, that it is self-defeating, and that it may well be unnecessary in the first place. As a result, the coloring strategy as a response to the normativity objection should be rejected, though it may have merits in intramural debates among naturalists. (shrink)

Abstract.We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [1], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.

We observe a number of connections between recent developments in the study of constraint satisfaction problems, irredundant axiomatisation and the study of topological quasivarieties. Several restricted forms of a conjecture of Clark, Davey, Jackson and Pitkethly are solved: for example we show that if, for a finite relational structure M, the class of M-colourable structures has no finite axiomatisation in first order logic, then there is no set (even infinite) of first order sentences characterising the continuously M-colourable structures amongst compact (...) totally disconnected relational structures. We also refute a rather old conjecture of Gorbunov by presenting a finite structure with an infinite irredundant quasi-identity basis. (shrink)

Is color real or illusory, mind independent or mind dependent? Does seeing in color give us a true picture of external reality? The metaphysical debate over color has gone on at least since the seventeenth century. In this book, M. Chirimuuta draws on contemporary perceptual science to address these questions. Her account integrates historical philosophical debates, contemporary work in the philosophy of color, and recent findings in neuroscience and vision science to propose a novel theory of the relationship between color (...) and physical reality. -/- Chirimuuta offers an overview of philosophy’s approach to the problem of color, finds the origins of much of the familiar conception of color in Aristotelian theories of perception, and describes the assumptions that have shaped contemporary philosophy of color. She then reviews recent work in perceptual science that challenges philosophers’ accounts of color experience. Finally, she offers a pragmatic alternative whereby perceptual states are understood primarily as action-guiding interactions between a perceiver and the environment. The fact that perceptual states are shaped in idiosyncratic ways by the needs and interests of the perceiver does not render the states illusory. Colors are perceiver-dependent properties, and yet our awareness of them does not mislead us about the world. Colors force us to reconsider what we mean by accurately presenting external reality, and, as this book demonstrates, thinking about color has important consequences for the philosophy of perception and, more generally, for the philosophy of mind. (shrink)

As standardly conceived, an illusion is an experience of an object o appearing F where o is not in fact F. Paradigm examples of color illusion, however, do not fit this pattern. A diagnosis of this uncovers different sense of appearance talk that is the basis of a dilemma for the standard conception. The dilemma is only a challenge. But if the challenge cannot be met, then any conception of experience, such as representationalism, that is committed to the standard conception (...) is false. Perhaps surprisingly, naïve realism provides a better account of color illusion.An apparence ymaad by som Magyk. Chaucer. (shrink)

Loosely put, colour constancy for example occurs when you experience a partly shadowed wall to be uniformly coloured, or experience your favourite shirt to be the same colour both with and without sunglasses on. Controversy ensues when one seeks to interpret ‘experience’ in these contexts, for evidence of a constant colour may be indicative a constant colour in the objective world, a judgement that a constant colour would be present were things thus and so, et cetera. My primary aim is (...) to articulate a viable conception of Present Constancy, of what occurs when a constant colour is present in experience, despite the additional presence of some experienced colour variation (e.g., correlating to a change in illumination). My proposed conception involves experienced colour layering – experiencing one opaque colour through another transparent one – and in particular requires one of those experienced layers to remain constant while the other changes. The aim is not to propose this layering conception of colour constancy as the correct interpretation of all constancy cases, but rather to develop the conception enough to demonstrate how it could and plausibly should be applied to various cases, and the virtues it has over rivals. Its virtues include a seamless application to constancy cases involving variations in filters (e.g., sunglasses) and illuminants; its ability to accommodate experiences of partial colours and error-free interpretations of difficult cases; and its broad theoretical-neutrality, allowing it to be incorporated into numerous perceptual epistemologies and ontologies. If layered constancy is prevalent, as I suspect it is, then our experiential access to colours is critically nuanced: we have been plunged into a world of colour without being told that we will rarely, if ever, look to a location and experience just one of them. (shrink)

The relations among consciousness, brain, behavior, and scientific explanation are explored in the domain of color perception. Current scientific knowledge about color similarity, color composition, dimensional structure, unique colors, and color categories is used to assess Locke.

Colours appear to instantiate a number of structural properties: for instance, they stand in distinctive relations of similarity and difference, and admit of a fundamental distinction into unique and binary. Accounting for these structural properties is often taken to present a serious problem for physicalist theories of colour. This paper argues that a prominent attempt by Byrne and Hilbert to account for the structural properties of the colours, consistent with the claim that colours are types of surface spectral reflectance, is (...) unsuccessful. Instead, it is suggested that a better account of the structural properties of the colours is provided by a form of non-reductive physicalism about colour: a naïve realist theory of colour, according to which colours are superficial mind-independent properties. (shrink)

In Color Harmonies, Augusto Garau systematically investigates the role of both color and form in visual perception and presents an original theory of the aesthetic relations among colors.

The chapter argues from an ethology-inspired internalist perspective that ‘colour’ is not a homogeneous and autonomous attribute, but rather plays different roles in different conceptual forms underlying perception. It discusses empirical and theoretical evidence that indicates that core assumptions underlying orthodox conceptions are grossly inadequate. The assumptions pertain to the idea that colour is a kind of autonomous and unitary attribute. It is regarded as unitary or homogeneous by assuming that its core properties do not depend on the type of (...) ‘perceptual object’ to which it pertains and that‘colour per se’ constitutes a natural attribute in the functional architecture of the perceptual system. It is regarded as autonomous by assuming that it can be studied in isolation of other perceptual attributes. More generally, the chapter argues on the bases of corresponding empirical and theoretical evidence that perception cannot be understood as the ‘recovery’ of physical world structure from sensory structure by input-based computational processes. Rather, the sensory input serves as a kind of sign for biologically relevant aspects of the external world that activates biologically given conceptual forms with their internal parameters. Although the sensory input is a causally necessary requirement for the activation of conceptual forms, the perceptual computations triggered are under the control of an internal programme based on a set of conceptual forms. (shrink)

I address the question of whether differences in color terminology cause differences in color experience in speakers of different languages. If linguistic representations directly affect color experience, then this is a case of what I call the linguistic penetrability of perception, which is a particular case of cognitive penetrability. I start with some general considerations about cognitive penetration and its alleged occurrence in the memory color effect. I then apply similar considerations to the interpretation of empirical studies of color perception (...) in speakers of different languages. I argue that findings such as differences in categorical perception in speakers of different languages do not show that language affects color experience. They therefore do not support the claim that color experience is linguistically penetrable. But even if we grant that color experience is different in speakers of different languages, I argue that this might still not be a case of linguistic penetration. Finally, I consider some epistemological consequences of the assumption that speakers of different languages have different color experiences. (shrink)

Colour has long been a source of fascination to both scientists and philosophers. In one sense, colours are in the mind of the beholder, in another sense they belong to the external world. Colours appear to lie on the boundary where we have divided the world into 'objective' and 'subjective' events. They represent, more than any other attribute of our visual experience, a place where both physical and mental properties are interwoven in an intimate and enigmatic way. -/- The last (...) few decades have brought fascinating changes in the way that we think about 'colour' and the role 'colour' plays in our perceptual architecture. In Colour Perception: Mind and the physical world, leading scholars from cognitive psychology, philosophy, neurophysiology, and computational vision provide an overview of the contemporary developments in our understanding of colours and of the relationship between the 'mental' and the 'physical'. With each chapter followed by critical commentaries, the volume presents a lively and accessible picture of the intellectual traditions which have shaped research into colour perception. -/- Written in a non-technical style and accessible to an interdisciplinary audience, the book will provide an invaluable resource for researchers in colour perception and the cognitive sciences. (shrink)

Are color adjectives ("red", "green", etc.) relative adjectives or absolute adjectives? Existing theories of the meaning of color adjectives attempt to answer that question using informal ("armchair") judgments. The informal judgments of theorists conflict: it has been proposed that color adjectives are absolute with standards anchored at the minimum degree on the scale, that they are absolute but have near-midpoint standards, and that they are relative. In this paper we report two experiments, one based on entailment patterns and one based (...) on presupposition accommodation, that investigate the meaning of scalar adjectives. We find evidence confirming the existence of subgroups of the population who operate with different standards for color adjectives. The evidence of interpersonal variation in where standards are located on the relevant scale and how those standards can be adjusted indicates that the existing theories of the meaning of color adjectives are at best only partially correct. We also find evidence that paradigmatic relative adjectives ("tall", "wide") behave in ways that are not predicted by the standard theory of scalar adjectives. We discuss several different possible explanations for this unexpected behavior. We conclude by discussing the relevance of our findings for philosophical debates about the nature and extent of semantically encoded context sensitivity in which color adjectives have played a key role. (shrink)