Connections Between BCK-algebras and Difference Posetse

We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.     

States on Polyadic MV-algebras

This paper is a contribution to the algebraic logic of probabilistic models of ?ukasiewicz predicate logic. We study the MV-states defined on polyadic MV-algebras and prove an algebraic many-valued version of Gaifman’s completeness theorem.     

Quasivarieties Generated by Simple MV-algebras

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.     

Frames of reference for the light-from-above prior in visual search and shape judgements

Faced with highly complex and ambiguous visual input, human observers must rely on prior knowledge and assumptions to efficiently determine the structure of their surroundings. One of these assumptions is the 'light-from-above' prior. In the absence of explicit light-source information, the visual system assumes that the light-source is roughly overhead. A simple, low-cost strategy would place this 'light-from-above' prior in a retinal frame of reference. A more complex, but optimal strategy would be to assume that the light-source is gravitationally up, and compensate for observer orientation. Evidence to support one or other strategy from psychophysics and neurophysiology has been mixed. This paper pits the gravitational and retinal frames against each other in two different visual tasks that relate to the light-from-above prior. In the first task, observers had to report the presence or absence of a target where distractors and target were defined purely by shading. In the second task, observers made explicit shape judgements of similar stimuli. The orientation of the stimuli varied across trials and the observer's head was fixed at 0, +/-45 or +/-60 degrees . In both tasks the retinal frame of reference dominated. Visual search behaviour with shape-from-shading stimuli (SFS) was modulated purely by stimulus orientation relative to the retina. However, the gravitational frame of reference had a significant effect on shape judgements, with a 30% correction for observer orientation. In other words, shading information is processed quite differently depending on the demands of the current task. When a 'quick and dirty' representation is required to drive fast, efficient search, that is what the visual system provides. In contrast, when the task is to explicitly estimate shape, extra processing to compensate for head orientation precedes the perceptual judgment. These results are consistent with current neurophysiological data on SFS if we re-frame compensation for observer orientation as a cue-combination problem.     

Duality for Double Quasioperator Algebras via their Canonical Extensions

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way. Presented by Robert Goldblatt     

The Dual Adjunction between MV-algebras and Tychonoff Spaces

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.     

Perfect MV-algebras are categorically equivalent to abelianl-groups

In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.Presented byDaniele Mundici     

Perspective taking in referring to objects behind versus in front of an observer: Frames of reference,intraindividual consistency,and response latencies

Speakers of English and German typically adopt the reflection variant of the relative frame of reference (FoR) in order to describe how nonoriented objects that are located in front of them are related to one another. Little is known, however, about how they proceed in dorsal settings, with objects located in their back. In this article, we explore the turn hypothesis, which assumes a (mental) 180° turn of the observer to face the objects, converting the dorsal into a frontal situation, so that the preferred FoR variant for frontal settings can be applied. To elicit spatial references, we used photographs that showed an observer and two objects either in the observer's visual field (frontal condition) or in the observer's back (dorsal condition). The observer was looking either in the same direction as the referencing individual (aligned perspectives) or in the opposite direction (vis-à-vis perspective). Data from two experiments show that while participants do adopt the observer's perspective, their references in dorsal settings are incompatible with the turn hypothesis. Analyses of response latencies indicate additional cognitive costs for establishing a FoR for the very first item in the dorsal condition as compared to the frontal condition, but fast adaption for subsequent items, and high intraindividual consistency in FoR choice in both conditions. Maintaining the assumption that references in dorsal settings should be compatible with the variant of the relative FoR adopted in frontal settings, participants’ references can be explained by assuming a backward projection that gets by without a (mental) turn of the observer.     

States on Pseudo MV-Algebras

Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.     

Generalized Kripke Frames

Algebraic work [9] shows that the deep theory of possible world semantics is available in the more general setting of substructural logics, at least in an algebraic guise. The question is whether it is also available in a relational form.This article seeks to set the stage for answering this question. Guided by the algebraic theory, but purely relationally we introduce a new type of frames. These structures generalize Kripke structures but are two-sorted, containing both worlds and co-worlds. These latter points may be viewed as modelling irreducible increases in information where worlds model irreducible decreases in information. Based on these structures, a purely model theoretic and uniform account of completeness for the implication-fusion fragment of various substructural logics is given. Completeness is obtained via a generalization of the standard canonical model construction in combination with correspondence results.The author’s research was partially supported by grant NSF01-4-21760 of the USA National Science Foundation as well as by a grant from the Carlsberg Foundation.Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen     

An Elementary Presentation of the Equivalence Between MV-algebras and l-groups With Strong Unit

Aim of this paper is to provide a self-contained presentation of the natural equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.     

The Intimate Relationship Between the McNaughton and the Chinese Remainder Theorems for MV-algebras

We show the intimate relationship between McNaughton Theorem and the Chinese Remaindner Theorem for MV-algebras. We develop a very short and simple proof of McNaughton Theorem. The arguing is elementary and right out of the definitions. We exhibit the theorem as just an instance of the Chinese theorem. Since the variety of MV-algebras is arithmetic, the Chinese theorem holds for MV-algebras. However, to make this paper self-contained and entirely elementary, we include a simple proof of this theorem inspired in Ferraioli and Lettieri (Math Logic Q 1:27–43, 2011).     

Fibring Argumentation Frames

This paper is part of a research program centered around argumentation networks and offering several research directions for argumentation networks, with a view of using such networks for integrating logics and network reasoning. In Section 1 we introduce our program manifesto. In Section 2 we motivate and show how to substitute one argumentation network as a node in another argumentation network. Substitution is a purely logical operation and doing it for networks, besides developing their theory further, also helps us see how to bring logic and networks closer together. Section 3 develops the formal properties of the new kind of network and Section 4 offers general discussion and comparison with the literature.