Interpolative realization of Boolean algebra frame for consistent treatment of gradation and/or fuzziness

This chapter presents a series questions in the philosophy of vagueness that will constitute the primary subjects of this book. The stance this book takes on these questions is outlined, and some preliminary ramifications are explored. These include the idea that (i) propositional vagueness is more fundamental than linguistic vagueness; (ii) propositions are not themselves sentence-like; they are coarse grained, and form a complete atomic Boolean algebra; (iii) vague propositions are, moreover, not simply linguistic constructions either such as sets of world-precisification pairs; and (iv) propositional vagueness is to be understood by its role in thought. Specific theses relating to the last idea include the thesis that one’s total evidence can be vague, and that there are vague propositions occupying every evidential role, that disagreements about the vague ultimately boil down to disagreements in the precise, and that one should not care intrinsically about vague matters.

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Assessing the Risk of Democratic Reversal in the United States: A Reply to Kurt Weyland

Political Science and Politics ◽

10.1017/s1049096521000329 ◽

2021 ◽

pp. 1-6

Author(s):

Matias López ◽

Juan Pablo Luna

Keyword(s):

United States ◽

Comparative Analysis ◽

Boolean Algebra ◽

Qualitative Comparative Analysis ◽

The United States ◽

American Democracy ◽

Institutional Dynamics ◽

The Public ◽

Original Analysis ◽

The Impact

ABSTRACT By replying to Kurt Weyland’s (2020) comparative study of populism, we revisit optimistic perspectives on the health of American democracy in light of existing evidence. Relying on a set-theoretical approach, Weyland concludes that populists succeed in subverting democracy only when institutional weakness and conjunctural misfortune are observed jointly in a polity, thereby conferring on the United States immunity to democratic reversal. We challenge this conclusion on two grounds. First, we argue that the focus on institutional dynamics neglects the impact of the structural conditions in which institutions are embedded, such as inequality, racial cleavages, and changing political attitudes among the public. Second, we claim that endogeneity, coding errors, and the (mis)use of Boolean algebra raise questions about the accuracy of the analysis and its conclusions. Although we are skeptical of crisp-set Qualitative Comparative Analysis as an adequate modeling choice, we replicate the original analysis and find that the paths toward democratic backsliding and continuity are both potentially compatible with the United States.

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Consistent treatment of rapidity divergence in soft-collinear effective theory

Journal of High Energy Physics ◽

10.1007/jhep03(2021)300 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Junegone Chay ◽

Chul Kim

Keyword(s):

Heavy Quark ◽

Soft Mode ◽

Form Factors ◽

Effective Theory ◽

Large Rapidity ◽

Soft Modes ◽

Heavy Quark Mass ◽

Soft Collinear Effective Theory ◽

Matching Procedure ◽

Consistent Treatment

Abstract In soft-collinear effective theory, we analyze the structure of rapidity divergence due to the collinear and soft modes residing in disparate phase spaces. The idea of an effective theory is applied to a system of collinear modes with large rapidity and soft modes with small rapidity. The large-rapidity (collinear) modes are integrated out to obtain the effective theory for the small-rapidity (soft) modes. The full SCET with the collinear and soft modes should be matched onto the soft theory at the rapidity boundary, and the matching procedure becomes exactly the zero-bin subtraction. The large-rapidity region is out of reach for the soft mode, which results in the rapidity divergence. The rapidity divergence in the collinear sector comes from the zero-bin subtraction, which ensures the cancellation of the rapidity divergences from the soft and collinear sectors. In order to treat the rapidity divergence, we construct the rapidity regulators consistently for all the modes. They are generalized by assigning independent rapidity scales for different collinear directions. The soft regulator incorporates the correct directional dependence when the innate collinear directions are not back-to-back, which is discussed in the N-jet operator. As an application, we consider the Sudakov form factor for the back-to-back collinear current and the soft-collinear current, where the soft rapidity regulator for a soft quark is developed. We extend the analysis to the boosted heavy quark sector and exploit the delicacy with the presence of the heavy quark mass. We present the resummed results of large logarithms in the form factors for various currents with the light and the heavy quarks, employing the renormalization group evolution on the renormalization and the rapidity scales.

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Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽

10.3390/axioms10030164 ◽

2021 ◽

Vol 10 (3) ◽

pp. 164

Author(s):

Songsong Dai

Keyword(s):

Boolean Algebra ◽

Orthomodular Lattice ◽

Orthomodular Lattices ◽

Approximation Operators ◽

Rough Approximation ◽

Distributive Law ◽

Complete Orthomodular Lattice ◽

The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

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Congruence pairs of principal MS-algebras and perfect extensions

Mathematica Slovaca ◽

10.1515/ms-2017-0430 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1275-1288

Author(s):

Abd El-Mohsen Badawy ◽

Miroslav Haviar ◽

Miroslav Ploščica

Keyword(s):

Boolean Algebra ◽

Congruence Lattices ◽

Descriptive Solution ◽

The One ◽

Special Case ◽

Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

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Programs as term transformers

Fundamenta Informaticae ◽

10.3233/fi-1980-3403 ◽

1980 ◽

Vol 3 (4) ◽

pp. 419-431

Author(s):

Stefan Sokołowski

Keyword(s):

Boolean Algebra ◽

Mathematical Study ◽

Partial Correctness

Predicates describing the states of computation may be regarded as functions into the Boolean algebra {false, true} and programs as transformers of those functions. If we do not restrict ourselves to this algebra, we get instead terms describing the states of computation and programs transforming the terms. In many cases this approach turns out to be more natural. This paper is a mathematical study of partial correctness and termination of programs in the language of term transformations.

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Information Retrieval Systems, an Algebraic Approach I1

Fundamenta Informaticae ◽

10.3233/fi-1981-4306 ◽

1981 ◽

Vol 4 (3) ◽

pp. 551-603

Author(s):

Zbigniew Raś

Keyword(s):

Information Retrieval ◽

Boolean Algebra ◽

Partially Ordered Set ◽

Ordered Set ◽

Algebraic Approach ◽

Partially Ordered ◽

Retrieval Systems ◽

Information Retrieval Systems ◽

Present Part ◽

L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

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CHOICE-FREE STONE DUALITY

Journal of Symbolic Logic ◽

10.1017/jsl.2019.11 ◽

2019 ◽

Vol 85 (1) ◽

pp. 109-148

Author(s):

NICK BEZHANISHVILI ◽

WESLEY H. HOLLIDAY

Keyword(s):

Boolean Algebra ◽

Boolean Algebras ◽

Stone Space ◽

Spectral Space ◽

Topological Representation ◽

Stone Duality ◽

Closed Sets ◽

Spectral Spaces ◽

Basic Facts ◽

Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

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The spectrum of partitions of a Boolean algebra

Archive for Mathematical Logic ◽

10.1007/s001530000065 ◽

2001 ◽

Vol 40 (4) ◽

pp. 243-254 ◽

Cited By ~ 2

Author(s):

J. Donald Monk

Keyword(s):

Boolean Algebra

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