On the Representation of N4-Lattices

N4-lattices provide algebraic semantics for the logic N4, the paraconsistent variant of Nelson's logic with strong negation. We obtain the representation of N4-lattices showing that the structure of an arbitrary N4-lattice is completely determined by a suitable implicative lattice with distinguished filter and ideal. We introduce also special filters on N4-lattices and prove that special filters are exactly kernels of homomorphisms. Criteria of embeddability and to be a homomorphic image are obtained for N4-lattices in terms of the above mentioned representation. Finally, subdirectly irreducible N4-lattices are described.     

Hoop twist-structures

In this paper, we introduce hoop twist-structure whose members are built as special squares of an arbitrary hoop. We show how our construction relates to eN4-lattices (N4-lattices) and implicative twist-structures. We prove that hoop twist-structures form a quasi-variety and characterize the AHT-congruences of each algebra in this quasi-variety in terms of the congruences of the associated hoop.     

Infinitary Action Logic: Complexity, Models and Grammars

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. Presented by Jacek Malinowski     

Brouwer-Zadeh logic,decidability and bimodal systems

We prove that Brouwer-Zadeh logic has the finite model property and therefore is decidable. Moreover, we present a bimodal system (BKB) which turns out to be characterized by the class of all Brouwer-Zadeh frames. Finally, we show that BrouwerZadeh logic can be translated into BKB.     

A semantical investigation on Brouwer-Zadeh logic

Conclusion In the standard approach to quantum mechanics, closed subspaces of a Hilbert space represent propositions. In the operational approach, closed subspaces are replaced by effects that represent a mathematical counterpart for properties which can be measured in a physical system. Effects are a proper generalization of closed subspaces. Effects determine a Brouwer-Zadeh poset which is not a lattice. However, such a poset can be embedded in a complete Brouwer-Zadeh lattice. From an intuitive point of view, one can say that these structures represent a natural logical abstraction from the structure of propositions of a quantum system. The logic that arises in this way is Brouwer-Zadeh logic. This paper shows that such a logic can be characterized by means of an algebraic and a Kripkean semantics. Finally, a strong completeness theorem for BZL is proved.     

BK-lattices. Algebraic Semantics for Belnapian Modal Logics

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK-lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK-lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK. Finally, we describe invariants determining a twist-structure over a modal algebra.     

Quantized intrinsically localized modes of a 3D lattice

Intrinsically Localized Modes are anharmonic oscillations found in one-dimensional systems, and occur relatively infrequently in classical higher dimensional lattices. However, when ILMs appear in simulations of classical lattices, their energies are too high for them to be seen in thermal equilibrium. We investigate quantized ILMs in a three-dimensional lattice using the Ladder Approximation, and find that ILMs occur preferentially for centre of mass momenta at which the van-Hove singularities in the two-phonon density of states coalesce. For interactions larger than a critical value, the ILMs form above the top of the two-phonon continuum, but fall into the continuum as q? is shifted away from the optimal value. This indicates that ILM excitations may be more ubiquitous in 3D lattices than previously expected. Furthermore, we find that the ILMs have intrinsic spins of either S = 0 or S = 2 and have internal structures associated with their spatial symmetry.     

Measure and integral with purely ordinal scales

We develop a purely ordinal model for aggregation functionals for lattice-valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modelling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 0 on the real interval [−1,1]. Mathematically we investigate the lattice of nonvoid intervals in a complete linear lattice, then the class of monotone interval-valued functions and their inner product.     

Fuzzy intuitionistic quantum logics

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL ³) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.     

On the Ranges of Algebraic Functions on Lattices

We study ranges of algebraic functions in lattices and in algebras, such as Łukasiewicz-Moisil algebras which are obtained by extending standard lattice signatures with unary operations.We characterize algebraic functions in such lattices having intervals as their ranges and we show that in Artinian or Noetherian lattices the requirement that every algebraic function has an interval as its range implies the distributivity of the lattice. Presented by Daniele Mundici     

Quantum MV algebras

We introduce the notion of quantum MV algebra (QMV algebra) as a generalization of MV algebras and we show that the class of all effects of any Hilbert space gives rise to an example of such a structure. We investigate some properties of QMV algebras and we prove that QMV algebras represent non-idempotent extensions of orthomodular lattices.I should like to thank Prof. M.L. Dalla Chiara and Dr. P. Minari for many interesting comments and remarks. Daniele Mundici     

Commodious Axiomatization of Quantifiers in Multiple-Valued Logic

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers.     

Priestley Duality for Paraconsistent Nelson’s Logic

The variety of N4^{^}{{\bf N4}^\perp}-lattices provides an algebraic semantics for the logic N4^{^}{{\bf N4}^\perp} , a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4^{^}{{\bf N4}^\perp}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.     

GROUPING BY PROXIMITY AND MULTISTABILITY IN DOT LATTICES:

Abstract— Gestalt phenomena have long resisted quantification. In the spirit of Gestalt field theory, we propose a theory that predicts the probability of grouping by proximity in the six kinds of dot lattices (hexagonal, rhombic, square, rectangular, centered rectangular, and oblique). We claim that the unstable perceptual organization of dot lattices is caused by competing forces that attract each dot to other dots in its neighborhood. We model the decline of these forces as a function of distance with an exponential decay function. This attraction function has one parameter, the attraction constant Simple assumptions allow us to predict the entropy of the perceptual organization of different dot lattices. We showed dot lattices tachistoscopically to 7 subjects, and from the probabilities of the perceived organisations, we calculated the entropy of each lattice for each subject. The model fit the data exceedingly well. The attraction constant did not vary much over subjects     

A Categorical Equivalence for Stonean Residuated Lattices

We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with other works and show some applications of the equivalence.

     

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators,and Applications to the Semantics of Non-Classical Logics I

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Canonical Extensions and Relational Representations of Lattices with Negation

This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations.We consider equational classes of lattices equipped with a negation operation ¬ which is dually self-adjoint (the pair (¬,¬) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, antilogism, pseudocomplementation or weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way.Among these classes, all but one—that of lattices with a negation which is an antilogism—were previously studied by W. Dzik, E. Or?owska and C. van Alten using Urquhart duality.In some cases in which a given axiom does not imply that negation is dually self-adjoint, canonicity is proven with the weaker assumption of antitonicity of the negation.     

Termination of logic programs with imperfect information: applications and query procedure

A general logic programming framework allowing for the combination of several adjoint lattices of truth-values is presented. The language is sorted, enabling the combination of several reasoning forms in the same knowledge base. The contribution of the paper is two-fold: on the one hand, sufficient conditions guaranteeing termination of all queries for the fix-point semantics for a wide class of sorted multi-adjoint logic programs are presented and related to some well-known probability-based formalisms; in addition, we specify a general non-deterministic tabulation goal-oriented query procedure for sorted multi-adjoint logic programs over complete lattices. We prove its soundness and completeness as well as independence of the selection ordering. We apply the termination results to probabilistic and fuzzy logic programming languages, enabling the use of the tabulation proof procedure for query answering.     

Quantum Logic in Intuitionistic Perspective

In their seminal paper Birkhoff and von Neumann revealed the following dilemma:[ ] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution, an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation, as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by Bruns and Lakser (1970) on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.