The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

Peters and Westerst?hl (Quantifiers in Language and Logic, 2006), and Westerst?hl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the ??classical?? Aristotelian squares of opposition and the ??modern?? Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, the vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( ${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$ , all or no) and Y ( ${\equiv{\rm I} \,{\wedge} \,{\rm O}}$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian.     

The Power of the Hexagon

The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name problem and it is not clear what is the intuitive notion corresponding to it. We explain then that the triangle of contrariety proposed by different people such as Vasiliev and Jespersen solves these problems, but that we don??t need to reject the square. It can be reconstructed from this triangle of contrariety, by considering a dual triangle of subcontrariety. This is the main idea of Blanché??s hexagon. We then give different examples of hexagons to show how this framework can be useful to conceptual analysis in many different fields such as economy, music, semiotics, identity theory, philosophy, metalogic and the metatheory of the hexagon itself. We finish by discussing the abstract structure of the hexagon and by showing how we can swing from sense to non-sense thinking with the hexagon.     

Violence Hexagon

In this article I will show why and how it is useful to exploit the hexagon of opposition to have a better and new understanding of the relationships between morality and violence and of fundamental axiological concepts. I will take advantage of the analysis provided in my book Understanding Violence. The Intertwining of Morality, Religion, and Violence: A Philosophical Stance. Springer, Heidelberg/Berlin, 2011) to stress some aspects of the relationship between morality and violence, also reworking some ideas by John Woods concerning the so-called epistemic bubbles, to reach and describe my own concept of moral bubbles. The study aims at providing a simple theory of basic concepts of moral philosophy, which extracts and clarifies the strict relationship between morality and violence and more, for example the new philosophical concept of overmorality. I will also conclude that this kind of hybrid diagrammatic reasoning is a remarkable example of manipulative explanatory abduction—through drawing—in the spirit of “conceptual structuralism”, promoted by Robert Blanché and further developed by Jean-Yves Béziau.     

The Sylvester-Chvatal Theorem

The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S on the line ab, or the line ab contains all the points in S. Chvatal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present article we prove this conjecture.     

Minimum Covering for Hexagon Triple Systems

A hexagon triple is a graph consisting of three triangles of the form (a, x, b), (b, y, c), and (c,z,a), where a, b, c, x, y, z are distinct. The triangle (a, b, c) is called the inside triangle and the triangles (a, x, b), (b,y,c), and (c, z, a) are called outside triangles. A 3k-fold hexagon triple system of order n is a pair (X, H), where H is an edge-disjoint collection of hexagon triples which partitions the edge set of 3kK ⁿ with vertex set X. Note that the outside triangles form a 3k-fold triple system. If the 3k-fold hexagon triple system (X, H) has the additional property that the inside triangles form a k-fold triple system, then (X, H) is said to be perfect. A covering of 3kK ⁿ with hexagon triples is a triple (X, H, P) such that: 1.3kK ⁿ has vertex set X. 2.P is a subset of E(λ K ⁿ ) with vertex set X for some λ, and 3.H is an edge disjoint partition of E(3kK ⁿ )∪ P with hexagon triples. If P is as small as possible (X, H, P) is called a minimum covering of 3kK ⁿ with hexagon triples. If the inside triangles of the hexagon triples in H form a minimum covering of kK ⁿ with triangles, the covering is said to be perfect. A complete solution for the problem of constructing perfect 3k-fold hexagon triple system and perfect maximum packing of 3kK ⁿ with hexagon triples was given recently by the authors [2]. In this work, we give a complete solution of the problem of constructing perfect minimum covering of 3kK ⁿ with hexagon triples.     

The Hyperplanes of the U 4(3) Near Hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U ₄(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative.     

Stone型定理

本文在Hilbert C~*-模框架下获得了Stone型定理,使得经典的Stone定理是其特例。     

The Simplest Tauberian Theorem

The following problem is considered: obtain the asymptotic properties of a function u from the asymptotic properties of the integral ₀ ^r u(t),dt. As is well known, this can be done under additional constraints on the function u(t). In this paper, we obtain a theorem in which these constraints are weaker than in other well-known versions of such theorems.     

The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.     

The Perron-Frobenius Theorem Revisited

An extension of the Perron-Frobenius Theorem is presented in the much more general setting of indecomposable semigroups of nonnegative matrices. Many features of the original theorem including the existence of a fixed positive vector, a block-monomial form, and spectral stability properties hold simultaneously for these semigroups. The paper is largely self-contained and the proofs are elementary. The classical theorem and some related results follow as corollaries.     

点是Gδ^－集的∑^＊－空间的构成定理

在林寿与我最近合作的一篇文章中指出了∑^*-空间的构成定理需重新考虑．本文就是要证明在空间X的每个点是Gδ^-集的条件下该构成定理是成立的，所得的结论是：X是T1且每个点是Gδ^-集的∑^ -空间，如果f：X→Y是闭的满连续映射，则在Y中有-σ-闭离散子空间Z，使得对每个y∈Y＼Z，f^-1(y)是X的ω1^-紧子空间．为得到该主要结果，本文证明了若空间X是每个点是Gδ^-集的次亚紧空间．则X中的每个闭离散子集是X中的Gδ^-集．     

高速取样定理

1948年Shannon给出了著名的适用于频谱有限函数的取样定理，从而使信号传输数字化成为可能，但是Shannon取样定理收敛慢等缺点已经不能满足通信技术的高速发展，本文在Shannon取样定理的基础上，选择新的取样定理函数，构造了具有收敛速度快，数值计算简单等优点的高速取样定理。     

Empty region graphs

A family of proximity graphs, called Empty Region Graphs (ERG) is presented. The vertices of an ERG are points in the plane, and two points are connected if their neighborhood, defined by a region, does not contain any other point. The region defining the neighborhood of two points is a parameter of the graph. This way of defining graphs is not new, and ERGs include several known proximity graphs such as Nearest Neighbor Graphs, β-Skeletons or Θ-Graphs. The main contribution is to provide insight and connections between the definition of ERG and the properties of the corresponding graphs.We give conditions on the region defining an ERG to ensure a number of properties that might be desirable in applications, such as planarity, connectivity, triangle-freeness, cycle-freeness, bipartiteness and bounded degree. These conditions take the form of what we call tight regions: maximal or minimal regions that a region must contain or be contained in to make the graph satisfy a given property. We show that every monotone property has at least one corresponding tight region; we discuss possibilities and limitations of this general model for constructing a graph from a point set.     

Approaching the Alethic Modal Hexagon of Opposition

Modal logic like many others sustains a hexagon of opposition, with the two ??additional?? vertices expressing contingency and non-contingency. We first illustrate hexagons of opposition generally by treating them as cut-down entailment lattices with order distinctions among multiple arguments suppressed. We then approach the modal case by treating it heuristically as a particular case of the hexagon for quantified propositions. Historically, possibility and contingency were sometimes confused: we show using the notion of duality that contingency, as negation-symmetric, is logically less interesting than possibility.     

Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in $\mathbb{R}^{d}$ , d≥3, in general position, that is, no (d+1) points of S lie in a common (d?1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3≤k≤d we provide a lower bound of $\varOmega(n^{d-k+1+2^{-d}})$ and strengthen this to Ω(n ^d?2/3) for k=2. On the way we provide various results on triangulations of point sets in  $\mathbb{R}^{d}$ . In particular, for any constant dimension d≥3, we prove that every set of n points (n sufficiently large), in general position in $\mathbb{R}^{d}$ , admits a triangulation with at least dn+Ω(logn) simplices.