Double Value-Ranges

2019 ◽  
pp. 167-181
Author(s):  
Peter Simons
Keyword(s):  
Combinatory Logic ◽  
Special Case ◽  
The Way

From the time of Begriffsschrift onwards, Frege treated functions of two or more places on a par with those of one place. This included the treatment of relations (Beziehungen) as a special case of polyadic functions in the way that concepts (Begriffe) were a special case of monadic functions. By the time of Grundgesetze (and unlike in Begriffsschrift), Frege dealt with relations largely through their extensions, which were what he called “double value-ranges” (Doppelwerthverläufe). This is in some ways a misnomer, since double value-ranges are simply a special case of single or ordinary value-ranges, namely value-ranges of functions derived from the value-ranges of monadic functions with additional saturated places. Frege’s treatment of the extensions of relations (which he came to call simply “Relationen”) thus embodies a move analogous to the treatment of polyadic functions as functions of functions, a device invented in 1920 by Moses Schönfinkel and since (unfairly) known in combinatory logic as “currying”. This paper considers the details of Frege’s Grundgesetze treatment of relations via their extensions, exhibits its grammar, and indicates its formal elegance by comparing it with other possible treatments.

scholarly journals Cubical Convex Ear Decompositions

10.37236/83 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Russ Woodroofe
Keyword(s):  
Finite Group ◽  
Necessary Conditions ◽  
Partition Lattice ◽  
Ear Decomposition ◽  
Interesting Special Case ◽  
Order Complexes ◽  
Special Case ◽  
The Way

We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a $CL$-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "$CL$-ced" or "$EL$-ced". We find an $EL$-ced of the $d$-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new $EL$-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets $P_{1}$ and $P_{2}$ have convex ear decompositions ($CL$-ceds), then their products $P_{1}\times P_{2}$, $P_{1}\check{\times} P_{2}$, and $P_{1}\hat{\times} P_{2}$ also have convex ear decompositions ($CL$-ceds). An interesting special case is: if $P_{1}$ and $P_{2}$ have polytopal order complexes, then so do their products.

Introduction: A New Way to Read Nature

2021 ◽  
pp. 1-10
Author(s):  
J. Arvid Ågren
Keyword(s):  
Nobel Prize ◽  
Nobel Prize Winner ◽  
Field Of Study ◽  
Natural Laws ◽  
The World ◽  
Living Organisms ◽  
Object Of Study ◽  
The Universe ◽  
Special Case ◽  
The Way

There really is something special about biology. The French biochemist and Nobel Prize winner Jacques Monod described its position among the sciences as simultaneously marginal and central (Monod 1970, p. xi). It is marginal, because its object of study—living organisms—are but a special case of chemistry and physics, contributing to only a minuscule part of the universe. Biology will never be the source of natural laws in the way physics is. At the same time, if, as Monod believed, the whole point of science is to understand humanity’s place in the world, then biology is the most central of them all. No other field of study deals so directly with the question of who we are and how we got here in the first place....

Weinberg's Refutation of Nominalism

Dialogue ◽  
1969 ◽  
Vol 8 (3) ◽  
pp. 460-474 ◽  
Author(s):  
Fred Wilson
Keyword(s):  
Fundamental Principle ◽  
General Argument ◽  
Special Case ◽  
The Way

Professor Weinberg, in his recent Abstraction, Relation, and Induction, has critically discussed the nominalistic tradition stemming from Ockham and continuing in the work of Berkeley and Hume. In this tradition there is one fundamental principle, which however divides into two parts. The first is (α) Whatever is distinguishable is distinct, and conversely. The second is (β) Whatever is distinct is separable, and conversely. Weinberg argues that both (α) and (β) are mistaken.In this paper I propose to explore the case against nominalism. I shall suggest that Weinberg's argument against (β), though not defective in the way some recent critics believe, depends upon a hidden premiss. I shall also suggest that the argument against (β), when the needed premiss is added, is but a special case of a more general argument. The latter in no way depends upon considerations concerning relational predicates, though Weinberg does in his discussion specifically introduce such considerations. Nor is that unreasonable on his part.

scholarly journals Petri nets based on Lawvere theories

2020 ◽  
Vol 30 (7) ◽  
pp. 833-864
Author(s):  
Jade Master
Keyword(s):  
Petri Nets ◽  
Operational Semantics ◽  
Lawvere Theory ◽  
Definition Of ◽  
Special Case ◽  
The Way

AbstractWe give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

scholarly journals Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph

2022 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Theoretical Study ◽  
Dominating Sets ◽  
Dominating Number ◽  
Coloring Number ◽  
Key Points ◽  
The Family ◽  
Different Types ◽  
Special Case ◽  
The Way

New setting is introduced to study “closing numbers” and “super-closing numbers” as optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. In this way, some approaches are applied to get some sets from (Neutrosophic)n-SuperHyperGraph and after that, some ideas are applied to get different types of super-closing numbers which are called by optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. The notion of dual is another new idea which is covered by these notions and results. In the setting of dual, the set of super-vertices is exchanged with the set of super-edges. Thus these results and definitions hold in the setting of dual. Setting of neutrosophic n-SuperHyperGraph is used to get some examples and solutions for two applications which are proposed. Both setting of SuperHyperGraph and neutrosophic n-SuperHyperGraph are simultaneously studied but the results are about the setting of n-SuperHyperGraphs. Setting of neutrosophic n-SuperHyperGraph get some examples where neutrosophic hypergraphs as special case of neutrosophic n-SuperHyperGraph are used. The clarifications use neutrosophic n-SuperHyperGraph and theoretical study is to use n-SuperHyperGraph but these results are also applicable into neutrosophic n-SuperHyperGraph. Special usage from different attributes of neutrosophic n-SuperHyperGraph are appropriate to have open ways to pursue this study. Different types of procedures including optimal-super-set, and optimal-super-number alongside study on the family of (neutrosophic)n-SuperHyperGraph are proposed in this way, some results are obtained. General classes of (neutrosophic)n-SuperHyperGraph are used to obtains these closing numbers and super-closing numbers and the representatives of the optimal-super-coloring sets, optimal-super-dominating sets and optimal-super-resolving sets. Using colors to assign to the super-vertices of n-SuperHyperGraph and characterizing optimal-super-resolving sets and optimal-super-dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on n-SuperHyperGraph to get new results about closing numbers and super-closing numbers alongside sets in the way that some closing numbers super-closing numbers get understandable perspective. Family of n-SuperHyperGraph are studied to investigate about the notions, super-resolving and super-coloring alongside super-dominating in n-SuperHyperGraph. In this way, sets of representatives of optimal-super-colors, optimal-super-resolving sets and optimal-super-dominating sets have key role. Optimal-super sets and optimal-super numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal-super ones. Simultaneously, three notions are applied into (neutrosophic)n-SuperHyperGraph to get sensible results about their structures. Basic familiarities with n-SuperHyperGraph theory and neutrosophic n-SuperHyperGraph theory are proposed for this article.

scholarly journals Gregory of Nyssa on the Metaphysics of the Trinity (with Reference to his Letter To Ablabius)

2018 ◽  
Author(s):  
Anna Marmodoro
Keyword(s):  
Philosophical Problem ◽  
Gregory Of Nyssa ◽  
The World ◽  
One And Many ◽  
The One ◽  
The Trinity ◽  
Special Case ◽  
The Way

This chapter explores Gregory’s metaphysics of the Trinity, which used an innovative distinction between stuffs (e.g. gold), which cannot be counted, and individuals (e.g. rings), which can. Gregory identifies the nature of any kind with the totality of its instances: the nature of man is the totality of men; the nature of gold is the totality of gold. For Gregory, the totality is more ‘real’ than the individuals into which it is articulated, which are merely the way in which the kind is present in the world. God is then identified as the total quantity of divinity in the world, and is thus one, and real. The Persons of the Trinity into which God is articulated are the ways God is in the world, and can be comprehended by us. Thus, the problem of the Trinity is solved as a special case of the philosophical problem of the One and Many.

On the role of implication in formal logic

2000 ◽  
Vol 65 (3) ◽  
pp. 1076-1114 ◽  
Author(s):  
Jonathan P. Seldin
Keyword(s):  
Higher Order ◽  
Formal Logic ◽  
Combinatory Logic ◽  
Cut Elimination ◽  
Classical Version ◽  
Logical Constants ◽  
Logical Strength ◽  
Special Case

AbstractEvidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a “classical” version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic.

scholarly journals Ewens Sampling and Invariable Generation

2018 ◽  
Vol 27 (6) ◽  
pp. 853-891 ◽  
Author(s):  
GERANDY BRITO ◽  
CHRISTOPHER FOWLER ◽  
MATTHEW JUNGE ◽  
AVI LEVY
Keyword(s):  
Symmetric Group ◽  
Fixed Number ◽  
Probability Measures ◽  
Random Permutations ◽  
Positive Probability ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Special Case ◽  
The Way

We study the number of random permutations needed to invariably generate the symmetric group Sn when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the special case α = 1 corresponds to uniformly random permutations.For strong α-logarithmic measures and almost every α, we show that precisely ⌈(1−αlog2)−1⌉ permutations are needed to invariably generate Sn with asymptotically positive probability. A corollary is that for many other probability measures on Sn no fixed number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Łuczak and Bovey to permutations obtained from the Ewens sampling formula.

How do novels hang together? Characterization as registerial meta-stability

2020 ◽  
Vol 40 (1) ◽  
pp. 29-48
Author(s):  
Fang Li ◽  
David Kellogg
Keyword(s):  
The Other ◽  
Middle Ground ◽  
The World ◽  
The One ◽  
Special Case ◽  
The Way

AbstractHow does a novel like Middlemarch cohere, since it is made up of at least two very different kinds of text, narrative on the one hand and dialogue on the other? In this paper, we look to two authorities: to literature, where authors seem to agree that it is consistency in voice that holds both narrators and characters together, and to linguistics, where a computerized corpus allows us to measure variation between and within characters. Where previous researchers found unsystematic variations, we find meta-stability: characters remain true to themselves only through variation. The way in which Dorothea addresses her future husband differs from the way she addresses her little sister in Chapter Five of Middlemarch but this is in turn a special case of differences between the way in which Dorothea addresses men and the way in which she addresses women. Such a difference serves to symbolically articulate a key theme of Eliot’s novel – the middle ground that every woman must occupy in the march from the world of our forefathers through that of our husbands to that of our children.

scholarly journals A Post-colonial Study of the Short Story “Araby” (1914) by James Joyce

2017 ◽  
Vol 8 (2) ◽  
pp. 201-208
Author(s):  
Pedram Maniee ◽  
Shahriyar Mansouri
Keyword(s):  
Short Stories ◽  
James Joyce ◽  
Short Story ◽  
Post Colonial ◽  
Clear Cut ◽  
Colonial Theory ◽  
Neighbor Country ◽  
Political Economic ◽  
Special Case ◽  
The Way

Abstract The short story of “Araby” by James Joyce was published in 1914 in Dubliners which is a collection of fifteen short stories set in the Dublin city of Northern Ireland. “Araby” is one of those short stories in which traces of the colonization of Ireland by the Great Britain in the nineteenth century can be found. Since the context of the short story is set in Dublin, analyzing it in light of post-colonial theory has made it a special case. Because despite the majority of literary works which are analyzed in light of post-colonial theory and in which the contrast between east and west geographically is quite visible, in “Araby” this contrast is not clear-cut and the culture of two neighbor countries are so close and as a consequent so difficult to claim cultural and religious colonization by a neighbor country. This essay investigates the way Joyce has portrayed the cultural, political, economic and social domination of Britain over Ireland, specifically Dublin. The essay also explores the context where Joyce had the motivation to write Dubliners and shows the fundamental principles of post-colonialism such as language, the notion of superior/inferior, cultural polyvalency, Self/Other and the critical tenets of Homi K. Bhabha including mimicry, liminality or hybridity and finds these tenets within this short story. The essay also investigates the way James Joyce has employed symbolism in order to portray his reaction to the domination of Britain over Ireland.

Sign in / Sign up

Export Citation Format

Share Document