We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa $, the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$. Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a (...) class='Hi'>subadditive strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$. We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $ -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa, \theta )$. We conclude the paper with an application to the failure of the infinite productivity of $\kappa $ -stationarily layered posets, answering a question of Cox. (shrink)

We prove that colouring of pairs from 2 with strong properties exists. The easiest to state problem it solves is: there are two topological spaces with cellularity 1 whose product has cellularity 2; equivalently, we can speak of cellularity of Boolean algebras or of Boolean algebras satisfying the 2-c.c. whose product fails the 2-c.c. We also deal more with guessing of clubs.

© The Authors 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: [email protected] Colour is a welcome work in history and philosophy of science.1 The opening chapters offer a fresh take on the history of perceptual theory and a broad overview of contemporary philosophy of colour. This is followed by the central fourth chapter, which introduces readers to a cluster of empirical data that to this point have not received explicit (...) treatment in philosophy. The core message is that instead of thinking of colour vision as filling in or painting the experienced world of primary qualities, we should think of colour vision as helping us to experience primary qualities. The colour visual system does not colour in an experiential world of primary qualities, it uses colour for the experience of those qualities. This colouring... (shrink)

The paper argues that colouring is a conventional ingredient of literal meaning characterized by a considerable degree of semantic under-determination and a high degree of context-sensitivity. The positive, though tentative, suggestion made in the paper is that whereas in the case of words such as "but" and "damn" we are dealing with words lacking in specificity, in the case of pejoratives in general, and racist jargon in particular, we are dealing with words that express concepts that purport to describe the (...) world as being in a certain way. The circumstance that in certain contexts of utterance colouring can be cancelled out, does not show that it forms a detachable part of a word's literal meaning. It only shows that to account for the interplay between context, literal meaning and assertoric content is much trickier than meets the eye. (shrink)

In this essay I consider Kaplan’s challenge to Frege’s so-called dictum: “Logic (and perhaps even truth) is immune to epithetical color”. I show that if it is to challenge anything, it rather challenges the view (attributable to Frege) that logic is immune to pejorative colour. This granted, I show that Kaplan’s inference-based challenge can be set even assuming that the pejorative doesn’t make any non-trivial truth-conditional (descriptive) contribution. This goes against the general tendency to consider the truth-conditionally inert logically irrelevant. (...) But I take it that Kaplan is right and take his examples to show that truth-conditional inertness need not entail inferential inertness. I end up assessing the Kaplan-Frege “debate” as giving edge to the former to the extent that clarity is achieved through Kaplanian inferences on what should be considered part of the explanandum. (shrink)

This article examines the early modern household's importance for producing experimental knowledge through an examination of the Halifax household of Margery and Henry Power. While Henry Power has been studied as a natural philosopher within the male-dominated intellectual circles of Cambridge and London, the epistemic labour of his wife, Margery Power, has hitherto been overlooked. From the 1650s, this couple worked in tandem to enhance their understanding of the vegetable world through various paper technologies, from books, paper slips and recipe (...) notebooks to Margery's drawing album and Henry's published Experimental Philosophy. Focusing on Margery's practice of hand-colouring flower books, her copied and original drawings of flowers and her experimental production of ink, we argue that Margery's sensibility towards colour was crucial to Henry's microscopic observations of plants. Even if Margery's sophisticated knowledge of plants never left the household, we argue that her contribution was nevertheless crucial to the observation and representation of plants within the community of experimental philosophy. In this way, our article highlights the importance of female artists within the history of scientific observation, the use of books and paperwork in the botanical disciplines, and the relationship between household science and experimental philosophy. (shrink)

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.

Colour shapes our world in profound, if sometimes subtle, ways. It helps us to classify, form opinions, and make aesthetic and emotional judgements. Colour operates in every culture as a symbol, a metaphor, and as part of an aesthetic system. Yet archaeologists have traditionally subordinated the study of colour to the form and material value of the objects they find and thereby overlook its impact on conceptual systems throughout human history.This book explores the means by which colour-based cultural understandings are (...) formed, and how they are used to sustain or alter social relations. From colour systems in the Mesolithic, to Mesoamerican symbolism and the use of colour in Roman Pompeii, this book paints a new picture of the past. Through their close observation of monuments and material culture, authors uncover the subtle role colour has played in the construction of past social identities and the expression of ancient beliefs. Providing an original contribution to our understanding of past worlds of meaning, this book will be essential reading for archaeologists, anthropologists and historians, as well as anyone with an interest in material culture, art and aesthetics. (shrink)

Colour plays a fundamental role in the philosophical treatments of painting. Colour while it is an essential part of the work of art cannot be divorced from the account of painting within which it is articulated. This paper begins with a discussion of the role of colour in Schelling's conception of art. Nonetheless its primary concern is to develop a critical encounter with Jean-François Lyotard's analysis of the Dutch painter Karel Appel. The limits of Lyotard's writings on painting, which this (...) paper will attribute in part to Lyotard's ‘empiricism’, becomes most apparent in his treatment of colour. (shrink)

A common view in both philosophy and the vision sciences is that, in human vision, wavelength information is primarily ‘for’ colouring: for seeing surfaces and various media as having colours. In this article we examine this assumption of ‘colour-for-colouring’. To motivate the need for an alternative theory, we begin with three major puzzles from neurophysiology, puzzles that are not explained by the standard theory. We then ask about the role of wavelength information in vision writ large. How might wavelength information (...) be used by any monochromat or dichromat and, finally, by a trichromatic primate with object vision? We suggest that there is no single ‘advantage’ to trichromaticity but a multiplicity, only one of which is the ability to see surfaces and so on as having categorical colours. Instead, the human trichromatic retina exemplifies a scheme for a general encoding of wavelength information given the constraints imposed by high spatial resolution object vision. Chromatic vision, like its partner, luminance vision, is primarily for seeing. Viewed this way, the ‘puzzles’ presented at the outset make perfect sense. 1 Reframing the Problem1.1 Introduction1.2 Three puzzles1.2.1 Why is trichromatic vision an anomaly in diurnal mammals?1.2.2 Why does the colour system occupy such a large and central part in human vision?1.2.3 Why are the blue cones so rare?1.3 Recasting the question2 The Costs and Benefits of Spectral Vision2.1 Spectral information and object vision2.2 Encoding the spectral dimension of light2.2.1 ‘General’ versus ‘specific’ encoding2.2.2 Spectral encoding and the monochromat2.2.3 Spectral encoding and the dichromat2.2.4 Spectral encoding and the human trichromat2.3 Three puzzles revisited2.3.1 Why is trichromatic vision an anomaly in diurnal mammals?2.3.2 Why does the colour system occupy such a large and central part in human vision?2.3.3 Why are the blue cones so rare3 Conclusion. (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)

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)

Summary Colourful plant images are often taken as the icon of natural history illustration. However, so far, little attention has been paid to the question of how this beautiful colouring was achieved. At a case study of the eighteenth-century Nuremberg doctor and botanist, Christoph Jacob Trew, the process of how illustrations were hand-coloured, who was involved in this work, and how the colouring was supervised and evaluated is reconstructed, mostly based on Trew's correspondence with the engraver and publisher of his (...) books, Johann Jacob Haid in Augsburg. Furthermore, the question of standardizing colours, their uses and their recipes is discussed at two examples of the same time period: the colour charts of the Bauer brothers, arguably the most renowned botanical draughtsmen of the period, and the colour tables by the Regensburg naturalist, Jacob Christian Schaeffer. Hand-colouring botanical illustrations, it is argued, was far from a straightforward task but confronted botanists and their employees with a plethora of practical and methodological problems, to which different solutions were developed in the course of time. Analysing these problems and solutions reveals some new and interesting aspects of the practices of eighteenth-century botany and of the production of scientific illustrations in general. (shrink)

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)

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)

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)

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.

We show that there exists a recursive coloring function c such that any $\Pi ^0_3$ set is not a solution to c for Hindman’s theorem. We also show that there exists a recursive coloring function c such that any $\Delta ^0_3$ set is not a solution to c for Hindman’s theorem restricted to sums of at most three numbers.

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.

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)

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.

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)

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)

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.

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.

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)