Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-based Algebras

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety ${\mathcal {A}}$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show that topological and discrete dualities for ${\mathcal {A}}$ exist in partnership.     

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.     

The Ackermann approach for modal logic,correspondence theory and second-order reduction

The problem of eliminating second-order quantification over predicate symbols is in general undecidable. Since an application of second-order quantifier elimination is correspondence theory in modal logic, understanding when second-order quantifier elimination methods succeed is an important problem that sheds light on the kinds of axioms that are equivalent to first-order correspondence properties and can be used to obtain complete axiomatizations for modal logics. This paper introduces a substitution-rewrite approach based on Ackermann?s Lemma to second-order quantifier elimination in modal logic. Compared to related approaches, the approach includes a number of enhancements: The quantified symbols that need to be eliminated can be flexibly specified. The inference rules are restricted by orderings compatible with the elimination order, which provides more control and reduces non-determinism in derivations thereby increasing the efficiency and success rate. The approach is equipped with a powerful notion of redundancy, allowing for the flexible definition of practical simplification and optimization techniques. We present correctness, termination and canonicity results, and consider two applications: (i) computing first-order frame correspondence properties for modal axioms and rules, and (ii) rewriting second-order modal problems to equivalent simpler forms. The approach allows us to define and characterize two new classes of formulae, which are elementary and canonical, and subsume the class of Sahlqvist formulae and the class of monadic inductive formulae.     

Natural Dualities Through Product Representations: Bilattices and Beyond

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation (not assumed to be an involution or commute with negation). Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras (so negation is absent).     

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.     

Finger posing primes number comprehension

Canonical finger postures, as used in counting, activate number knowledge, but the exact mechanism for this priming effect is unclear. Here we dissociated effects of visual versus motor priming of number concepts. In Experiment 1, participants were exposed either to pictures of canonical finger postures (visual priming) or actively produced the same finger postures (motor priming) and then used foot responses to rapidly classify auditory numbers (targets) as smaller or larger than 5. Classification times revealed that manually adopted but not visually perceived postures primed magnitude classifications. Experiment 2 obtained motor priming of number processing through finger postures also with vocal responses. Priming only occurred through canonical and not through non-canonical finger postures. Together, these results provide clear evidence for motor priming of number knowledge. Relative contributions of vision and action for embodied numerical cognition and the importance of canonicity of postures are discussed.     

WD 40: Historical Insights of Ellen White and John Dewey into Three Dualistic Concerns: Implications for Current Practice

Abstract

The dualisms between theory and practice, individual and society, intellectual and manual, as identified by John Dewey in the 19th century, find their direct correspondence in the writings of a contemporary Christian writer of his era, Ellen G. White, whose publications and life work led to the development of what is now one of the single largest Protestant private education systems in the world. This paper provides an outline of these three historical dualities, identifies the shared insights of Ellen White and John Dewey relating to these dualities within their historical context, and delineates the implications for current practice.     

Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality.In part III we discuss applications in logic of the framework developed. Specifically, logics with restricted structural rules give rise to lattices with normal operators (in our sense), such as the Full Lambek algebras (F L-algebras) studied by Ono in [36]. Our Stone-type representation results can be then used to obtain canonical constructions of Kripke frames for such systems, and to prove a duality of algebraic and Kripke semantics for such logics.     

JOHANNINE DUALISM AND CONTEMPORARY PLURALISM

The main thrust of this essay is an argument that (1) John's Gospel is not “dualistic” in the technical sense of the term but that it does operate with “oppositional dualities”; (2) far from being negating of pluralism as both a social fact and a political project, John's kind of oppositional dualities may in fact be the best way to preserve social pluralism and make it flourish; (3) two notable features of John's oppositional dualities are that (i) they are soft enough to allow surprisingly blurry boundaries between “inside” and “outside” and that (ii) they mandate a willingness to sacrifice one's own life on behalf of one's deadly enemies.     

The role of vision in the development of finger-number interactions: Finger-counting and finger-montring in blind children

Previous research has suggested that the use of the fingers may play a functional role in the development of a mature counting system. However, the role of developmental vision in the elaboration of a finger numeral representation remains unexplored. In the current study, 14 congenitally blind children and 14 matched sighted controls undertook three different test batteries that examined (a) general cognitive abilities, (b) the spontaneous use of finger-counting and finger-montring strategies (where “finger-montring” is a term used to characterize the way people raise their fingers to show numerosities to other people), and (c) the canonicity level of the finger-counting and finger-montring habits. Compared with sighted controls, blind children used their fingers less spontaneously to count and in a less canonical way to count and show quantities. These results demonstrate that the absence of vision precludes the development of a typical finger numeral representation and suggest that the use of canonical finger-counting and finger-montring strategies relies on the visual recognition of particular hand shapes.     

Optimal natural dualities for varieties of Heyting algebras

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.     

Book Reviews

Interpreting the diamond of modal logic as the derivative, we present a topological canonical model for extensions of K4 and show completeness for various logics. We also show that if a logic is topologically canonical, then it is relationally canonical.     

Influence in canonical correlation analysis

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.We thank the Editor and the anonymous reviewers for their helpful comments. This research was carried out with the financial support of the Italian Ministry of the University and the National Research Council.     

Regularized Multiple-Set Canonical Correlation Analysis

Multiple-set canonical correlation analysis (Generalized CANO or GCANO for short) is an important technique because it subsumes a number of interesting multivariate data analysis techniques as special cases. More recently, it has also been recognized as an important technique for integrating information from multiple sources. In this paper, we present a simple regularization technique for GCANO and demonstrate its usefulness. Regularization is deemed important as a way of supplementing insufficient data by prior knowledge, and/or of incorporating certain desirable properties in the estimates of parameters in the model. Implications of regularized GCANO for multiple correspondence analysis are also discussed. Examples are given to illustrate the use of the proposed technique. The work reported in this paper is supported by Grants 10630 and 290439 from the Natural Sciences and Engineering Research Council of Canada to the first and the second authors, respectively. The authors would like to thank the two editors (old and new), the associate editor, and four anonymous reviewers for their insightful comments on earlier versions of this paper. Matlab programs that carried out the computations reported in the paper are available upon request.     

Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

This paper presents an algebraic approach of some many-valued generalizations of modal logic. The starting point is the definition of the [0, 1]-valued Kripke models, where [0, 1] denotes the well known MV-algebra. Two types of structures are used to define validity of formulas: the class of frames and the class of ? _n -valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of ? _n ) of the allowed truth values of the formulas in u. We apply and develop algebraic tools (namely, canonical and strong canonical extensions) to generate complete modal n + 1-valued logics and we obtain many-valued counterparts of Shalqvist canonicity result.     

Diachronic and synchronic unity

There are two different varieties of question concerning the unity of consciousness: questions about unity at a time, and unity over time. A recent trend in the debate about unity has been to attempt to provide a ‘generalized’ account that purports to solve both problems in the same way. This attempt can be seen in the accounts of Barry Dainton and Michael Tye. In this paper, I argue that there are crucial differences between unity over time and unity at a time that make it impossible to provide a generalized account of unity. The source of these crucial differences is the phenomenon of the ‘continuity of consciousness’. I argue that accounts of unity over time have to provide an account of this continuity, and that there is no phenomenon analogous to continuity in the case of unity at a time. Attention to the continuity of consciousness reveals crucial structural differences between the two varieties of unity. These structural differences make it impossible to provide a generalized account of unity. I show that the problems faced by Dainton’s and Tye’s accounts in the light of the structural differences make their accounts of unity appear far less appealing than they might initially have looked. I conclude by noting that, in the light of the important differences between the two varieties of unity, it is a mistake to attempt to model accounts of unity over time on accounts of unity at a time.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

Scaffolding within the structure of Dialogical Self: Hierarchical dynamics of semiotic mediation

Scaffolding is a notion that allows us to conceptualize direction towards change. As a form of guidance, scaffolding may result in both change and non-change. In this paper I apply the notion of scaffolding by signs (semiotic mediation) to the theory of Dialogical Self (DS). The DS is a construct that brings into psychology a new way of theoretical thought—thinking in dualities. Dualities are systemic units of two opposites that are mutually related by functional dynamic relations. Within the theory of DS, human psychological functioning is explained by transformations of constantly changed I-positions that are mapped both structurally (internal/external) and temporally (past/present/future). Semiotic mediation within the DS guarantees the person's psychological distancing from the here-and-now setting. This distancing is guided by promoter signs—generalized meanings of field-like form that orient the self's transformation. These signs are parts of the semiotic mediating processes where higher-level signs guide the range of openness of the sign hierarchy itself for further transformation when that is needed.