Studies in the axiomatic foundations of Boolean algebra. I.

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.

Download Full-text

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.

Download Full-text

Sessions of the Seminar “Algebra i Logika”

Algebra and Logic ◽

10.1007/s10469-021-09616-0 ◽

2021 ◽

Vol 59 (6) ◽

pp. 493-493

Keyword(s):

Algebra I

Download Full-text

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.

Download Full-text

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.

Download Full-text