
Authors:
Janis CIRULIS
Title:Simple Logics for Basic Algebras
Pages:95110
File:bibtex
Abstract ( + )
An MValgebra is an algebra (A,⊕,┐,0), where (A,⊕,0) is a commutative monoid and ┐ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also noncommutative) generalizations of MValgebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbertstyle systems, with implication and negation as the primitive connectives. We present a considerably simpler logic, L_{B}, for basic algebras, where implication and falsum are taken as primitives. We also consider some subvarieties of basic algebras known in the literature, discuss classes of implicational algebras termequivalent to each of these varieties, and construct axiomatic extensions of L_{B} for which these classes serve as equivalent algebraic semantics.

Authors:
Zofia KOSTRZYCKA
Title:On Halldén Completeness of Modal Logics Determined By Homogeneous Kripke Frames
Pages:111130
File:bibtex
Abstract ( + )
Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.

Authors:
Feng GAO and George TOURLAKIS
Title:A Short and Readable Proof of Cut Elimination for Two FirstOrder Modal Logics
Pages:131148
File:bibtex
Abstract ( + )
A well established technique toward developing the proof theory of a Hilbertstyle modal logic is to introduce a Gentzenstyle equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1],[6],[8],[18],[10],[12]). In the firstorder modal case, on one hand we know that the Gentzenisation of the straightforward firstorder extension of GL, the logic QGL, admits no cut elimination (if the rule is included as primitive; or, if not included, then the rule is not admissible [1]). On the other hand the (cutfree) Gentzenisations of the firstorder modal logics M_{3} and ML_{3} of [10],[12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M_{3} and ML_{3}, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cutfree Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M_{3} and M_{3}, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules.

Authors:
Alexej P. PYNKO
Title:Minimal Sequent Calculi for Lukasiewicz’s FinitelyValued Logics
Pages:149154
File:bibtex
Abstract ( + )
The primary objective of this paper, which is an addendum to the author's [8], is to apply the general study of the latter to Lukasiewicz's nvalued logics [4]. The paper provides an analytical expression of a (n1)place sequent calculus (in the sense of [5]) with the cutelimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi in the sense of a complexity of systems of premises of introduction rules. This together with a quite effective procedure of construction of an equality determinant\/ (in the sense of [5]) for the logics involved to be extracted from the constructive proof of Proposition 6.10 of [6] yields an equally effective procedure of construction of both Gentzenstyle [2](i.e., 2place ) and Taitstyle [11] (i.e., 1place) minimal sequent calculi following the method of translations described in Subsection 4.2 of [7].

Authors:
A. V. FIGALLO and G. PELAITAY
Title:Tense Polyadic n⨯ m Valued ŁukasiewiczMoisil Algebras
Pages:155181
File:bibtex
Abstract ( + )
In 2015, A.V. Figallo and G. Pelaitay introduced tense n× mvalued ŁukasiewiczMoisil algebras, as a common generalization of tense Boolean algebras and tense nvalued ŁukasiewiczMoisil algebras. Here we initiate an investigation into the class tpLM_{n× m} of tense polyadic n× mvalued ŁukasiewiczMoisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic nvalued ŁukasiewiczMoisil algebras studied by Chiriţă in 2012. Our main result is a representation theorem for tense polyadic n× mvalued ŁukasiewiczMoisil algebras.