Bulletin of the Section of Logic

Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \(\left(A,f,g\right)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)\leq b\) iff \(a\leq g(b)\), and \(g(a\rightarrow b)\leq g(a)\rightarrow g(b)\) forall \(a,b\in A\). In this paper, we are going to prove that some varieties of HilGC-algebras are characterized by first-order conditions defined in the dual space and that these varieties are canonical. Additionally, we will also study and characterize the congruences of an HilGC-algebra through specific closed subsets of the dual space. This characterization will be applied to determine the simple algebras and subdirectly irreducible HilGC-algebras.

Sergio A. Celani

Daniela Montagie

Hilbert Algebras with Hilbert-Galois Connections II

希尔伯特代数与希尔伯特-伽罗瓦连接 II

Ulrich (2005) showed that most of the known axiomatisations of the classical equivalence calculus (EC) are D-incomplete, that is, they are not complete with the condensed detachment rule (D) as the primary rule of the proof procedure. He proved that the axiomatisation \(\{EEpEqrErEqp, EEEpppp\}\) by Wajsberg (1937) is D-complete and pointed out a number of D-complete single axioms, including one organic single axiom. In this paper we present new single axioms for EC with the condensed detachment and the reversed condensed detachment rules that form D-complete bases and are organic.

Marcin Czakon

D-complete Single Axioms for the Equivalential Calculus with the rules D and R

D和R规则下等值演算的D-完备单公理

In this manuscript, we have presented the concept of \(\mathcal{L}\)-weakly 1-absorbing prime ideals and \(\mathcal{L}\)-weakly 1-absorbing prime filters within an ADL. Mainly, we illustrate the connections between \(\mathcal{L}\)-weakly prime ideals (filters) and \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters), as well as between \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) and \(\mathcal{L}\)-weakly 2-absorbing ideals (filters). Lastly, we have shown that both the image and inverse image of \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) result in \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters).

Natnael Teshale Amare

\(\mathcal{L}\)−weakly 1−Absorbing Prime Ideals and Filters

\(\mathcal{L}\)−弱1−吸收素理想与滤子

When humans reason, they are able to revise their beliefs in light of new information and abandon obsolete conclusions. Logicians argued, that in some cases, such reasonings appear to be non-monotonic. Thus, many different, seemingly non-monotonic systems were created to formally model such cases. The purpose of this article is to re-examine the definition of non-monotonicity and its implementation in non-monotonic logics and in examples of everyday human reasoning. We will argue that many non-monotonic logics employ some weakened versions of the definitions of non-monotonicity, since in-between different steps of reasoning they either: a) allow previously accepted premises to be removed, or b) change the rules of inference. Of the two strategies, the second one seems downright absurd, since changing the rules of a given logic is a mere replacement of that logic with the rules of another. As a consequence we obtain two logics, whereas the definition of a non-monotonic logic is supposed to define one. The definition of non-monotonicity does not permit either of these cases, which means that such logics are monotonic.

Piotr Łukowski

Konrad Rudnicki

Exploring the Definition of Non-Monotonicity – Logical and Psychological Considerations

探索非单调性的定义——逻辑与心理学的考量

When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop.

Rajab Ali Borzooei

Elham Babaei

Constructing a Hoop Using Rough Filters

用粗糙滤波器构建环

The paper will study extensions of t-normal logics S0.5o and S0.5 using some axioms of normal logics. We will prove determination theorems for these extensions by appropriate Kripke-style models. It will allow us to obtain the determinations of the logics K45, KB4 (= KB5), KD45 and S5 without using modal rules.

ISSN	0138-0680, 2449-836X
h-index	16

Publication Information	Publisher: Department of Logic, University of Lodz，Publishing cycle: ，Journal Type: journal，Open Access Journals: Yes
Basic data	Year of publication: 1996，Proportion of original research papers: 0.00%，Self Citation Rate:N.A.， Gold OA Rate: 0.00%
Average review cycle	网友分享经验：8 Weeks

Bulletin of the Section of Logic

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Hilbert Algebras with Hilbert-Galois Connections II

D-complete Single Axioms for the Equivalential Calculus with the rules D and R

\(\mathcal{L}\)−weakly 1−Absorbing Prime Ideals and Filters

Exploring the Definition of Non-Monotonicity – Logical and Psychological Considerations

Constructing a Hoop Using Rough Filters

Hilbert Algebras with Hilbert-Galois Connections II

\(\mathcal{L}\)−weakly 1−Absorbing Prime Ideals and Filters

D-complete Single Axioms for the Equivalential Calculus with the rules D and R

Exploring the Definition of Non-Monotonicity – Logical and Psychological Considerations

Some additional axioms for t-normal logics. Defining K45, KB4, KD45 and S5 without using modal rules