Abstract
In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second ...we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal filters of quasi-Boolean algebras, showing that the prime filters of a quasi-Boolean algebra are precisely the maximal filters and the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space. Finally, we define and study the reticulation of a quasi-Boolean algebra.
We study the interplay between the properties of measures on a Boolean algebra A and the forcing names for ultrafilters on A. We show that several well known measure theoretic properties of Boolean ...algebras (such as supporting a strictly positive measure or carrying only separable measures) have quite natural characterizations in the forcing language. We show some applications of this approach. In particular, we reprove a theorem of Kunen saying that in the classical random model there are no towers of height ω2.
In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that ...counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.
This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa). According to Rudolf Wille, a double Boolean algebra $\underline{D}:=(D, \sqcap, ...\sqcup, \neg, \lrcorner, \bot, \top)$ is an algebra of type $(2, 2, 1, 1, 0, 0),$ which satisfies a set of properties. This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts. We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures. We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra $D_{\sqcup}.$
In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice ...order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.
There are some methods of proof of the compactness theorem for classical logic which bypass the completeness theorem. Among them are the purely topological one, the purely algebraic one, and the ...hybrid one. These methods make essential use of either Tychonoff's Theorem, the concept of ultraproducts or the concept of Cantor sets as topological spaces. Instead of these conceptual tools, the paper provides the theorem with a method of proof that appeals to the concept of Stone spaces of Boolean algebras. In connection with a classical logical system (a propositional calculus or a predicate calculus), the method consists of five components. Firstly, the problem of the compactness of the logical system is reduced to that of the compactness of some topological space. Secondly, what is called the Lindenbaum algebra of the system is set up, which is in fact a Boolean algebra. Thirdly, it has to be shown that the Stone space of the Boolean algebra is compact. Fourthly, the set of sentences whose equivalent classes are members of the Stone space is shown to be satisfiable or simultaneously true. Finally, a homeomorphism is constructed between the topological space and the compact Stone space. Additionally, the method admits of a natural generalisation to the proof of the compactness theorem for modal logic.
This work presents an interactive proof assistant, based on Dijkstra-Scholten logic, aimed at teaching logic and discrete mathematics in higher education. The assistant interface is web and easy to ...use, since inferences can be made just with the mouse. The educational experience is presented showing a correlation between the grades of the assessments in class and those made with the application web. Additionally, an algorithm proof theory for the Disjktra-Scholten system are made and the following algorithms are shown: 1) a versatile printing algorithm that allows the administrator to configure the symbols of a theory, by assigning the desired presentation with LaTeX; 2) An algorithm, based on Broda and Damas combinators, for generate monotonic or anti monotonic inferences in the Dijkstra-Scholten logic; 3) An algorithm to generate the proofs of dual theorems in Boolean Algebra theory.
We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal ...characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.
We propose a general system that combines the powerful features of modal logic and many-sorted reasoning. Its algebraic semantics leads to a many-sorted generalization of boolean algebras with ...operators, for which we prove the analogue of the Jónsson-Tarski theorem. Our goal was to deepen the connections between modal logic and program verification, while also testing the expressiveness of our system by defining a small imperative language and its operational semantics.
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