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)

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)

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.

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 problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that (...) there is an effective degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring. (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)

Studies to neutrosophic graphs happens to be not only innovative and interesting, but gives a new dimension to graph theory. The classic coloring of edge problem happens to give various results. Neutrosophic tree will certainly find lots of applications in data mining when certain levels of indeterminacy is involved in the problem. Several open problems are suggested.

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotation theories^ can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, (...) including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21]. (shrink)

The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi-gated.

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and (...) culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

Special issue in honour of Landon Rabern. This special issue of Discrete Mathematics is dedicated to his memory, as a tribute to his many research achievements. It contains 10 new articles written by his collaborators, friends, and colleagues that showcase his interests.

We analyze a solitaire game in which a demon rearranges some cards after each move. The graph edge coloring theorems of K˝onig (1931) and Vizing (1964) follow from the winning strategies developed.

Many stochastic search algorithms have recently been developed to make more remarkable progress than systematic search algorithms because stochastic algorithms sometimes solve large-scale constraint satisfaction problems in a practical time. However, such stochastic algorithms have the drawback of getting stuck in local optima which are not acceptable as final solutions. We analyze an iterative improvement algorithm from the viewpoint of constraint structures causing local optima. Using the graph-coloring problem with three colors, an archetype problem to evaluate constraint satisfaction (...) algorithms, we study the local graph structures around which so many conflicts thrash. We propose a key constraint structure, an LM pair, which may induce a local optimum, and clarify the mechanism of how conflicting colors for an LM pair obstruct the stepwise refinement of hill-climbing. Experimental results show that LM pairs are strongly correlated with the search efficiency of the stochastic search algorithm. (shrink)

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed (...) class='Hi'>graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.We refute various reasonable generalizations of these results to hypergraphs. (shrink)

Paraconsistent logic is an area of philosophical logic that has yet to find acceptance from a wider audience. The area remains, in a word, disreputable. In this essay, we try to reassure potential consumers that it is not necessary to become a radical in order to use paraconsistent logic. According to the radicals, the problem is the absurd classical account of contradiction: Classically inconsistent sets explode only because bourgeois classical semantics holds, in the face of overwhelming evidence to the contrary, (...) that both A and ∼ A cannot simultaneously be true! We suggest (more modestly) that there is, at least sometimes, something else worth preserving, even in an inconsistent, unsatisfiable premise set. In this paper we present, in a new guise, a very general version of this “preservationist” approach to paraconsistency. (shrink)

We prove that the statement “there is aksuch that for everyfthere is ak-bounded diagonally nonrecursive function relative tof” does not imply weak König’s lemma over${\rm{RC}}{{\rm{A}}_0} + {\rm{B\Sigma }}_2^0$. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that everyk-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of${\rm{I\Sigma }}_2^0$.

We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $ -regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $ -regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $ -regular forest. We also show that if such a problem admits a computable solution on every computable maximum (...) degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold. (shrink)

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using interpolation we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound (...) on interpolants in this fragment.We then apply these results to extend and improve previous results on multilinear proofs , as studied in [Ran Raz, Iddo Tzameret, The strength of multilinear proofs. Comput. Complexity ]. Specifically, we show the following:• Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations.• Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the functional pigeonhole principle. The latter are different from previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system. (shrink)

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the (...) least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs. (shrink)

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max, which satisfy [Formula: see text] and prove: Theorem. For every Polish space X and every continuous (...) pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row. (shrink)

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. (...) Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)

We consider three problems concerning alpha conversion of closed terms (combinators). (1) Given a combinator M find the an alpha convert of M with a smallest number of distinct variables. (2) Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N. (3) Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way. We obtain the following results. (...) (1) There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs. (2) Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2). (3) At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given. There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand. (shrink)

The problem of the existence of a universal structure omitting a finite set of forbidden substructures is reducible to the corresponding problem in the category of graphs with a vertex coloring by two colors. It is not known whether this problem reduces further to the category of ordinary graphs. It is also not known whether these problems are decidable.

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 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)

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)

© 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)

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)

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.

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)

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)

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 introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook and Rabern, Rabern, and Macauley. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It (...) is shown that this notion of paradoxicality coincides with that of Kripke. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs. (shrink)

This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.