Free Q-distributive lattices

The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.     

Distributive lattices with an operator

It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences.I would like to thank my research supervisor Dr. Roberto Cignoli for his helpful suggestions during the preparation of this paper and the referee for calling my attention to Goldblatt's paper [5].     

Categorizing Smells: A Localist Approach

Humans are poorer at identifying smells and communicating about them, compared to other sensory domains. They also cannot easily organize odor sensations in a general conceptual space, where geometric distance could represent how similar or different all odors are. These two generalities are more or less accepted by psychologists, and they are often seen as connected: If there is no conceptual space for odors, then olfactory identification should indeed be poor. We propose here an important revision to this conclusion: We believe that the claim that there is no odor space is true only if by odor space, one means a conceptual space representing all possible odor sensations, in the paradigmatic sense used for instance for color. However, in a less paradigmatic sense, local conceptual spaces representing a given subset of odors do exist. Thus the absence of a global odor space does not warrant the conclusion that there is no olfactory conceptual map at all. Here we show how a localist account provides a new interpretation of experts and cross-cultural categorization studies: Rather than being exceptions to the poor olfactory identification and communication usually seen elsewhere, experts and cross-cultural categorization are here taken to corroborate the existence of local conceptual spaces.     

Typicality and Graded Membership in Dimensional Adjectives

This paper concerns an investigation of the manner in which typicality constrains graded membership in antonymous dimensional adjectives such as short/tall and cheap/expensive using the conceptual spaces framework. In this framework, items are organized in a space comprised of one or more dimensions along which they can be compared. The items’ graded membership is established by their relative proximity in this space to the prototypical instances of contrasting concepts. Because dimensional adjectives can be applied to an indefinite variety of things and grammatically have no upper bound to serve as cognitive reference point, they have been argued to lack prototypes. We present the results of an empirical study showing that the conceptual spaces framework can nevertheless be extended successfully to dimensional adjectives by complementing them with a comparison class argument (such as short/tall for an adult man and cheap/expensive for a smartphone), allowing participants to retrieve meaningful prototypical instances, which can be used to establish membership degree. Since dimensional adjectives are subjective, we investigate how the framework can accommodate interindividual variability in membership degree judgments. We find that the predictions of the framework significantly improve if prototypical instances themselves are assumed to come with a gradient instead of being considered equally typical, thereby providing a more fine‐grained account of typicality and furthering the development of the conceptual spaces framework.     

The Finite-Dimensional Freeman Thesis

I suggest a modification—and mathematization—of Freeman’s thesis on the relations among “perception”, “the finite brain”, and “the world”, based on my recent proposal that the theory of finite topological spaces is both an adequate and a natural mathematical foundation for human psychology.

How Category Selection Impacts Inference Reliability: Inheritance Inference From an Ecological Perspective

This article presents results from a simulation-based study of inheritance inference, that is, inference from the typicality of a property among a “base” class to its typicality among a subclass of the class. The study aims to ascertain which kinds of inheritance inferences are reliable, with attention to the dependence of their reliability upon the type of environment in which inferences are made. For example, the study addresses whether inheritance inference is reliable in the case of “exceptional subclasses” (i.e., subclasses that are known to be atypical in some respect) and attends to variations in reliability that result from variations in the entropy level of the environment. A further goal of the study is to show that the reliability of inheritance inference depends crucially on which sorts of base classes are used in making inferences. One approach to inheritance inference treats the extension of any atomic predicate as a suitable base class. A second approach identifies suitable base classes with the cells of a partition (of a preselected size k) of the domain of objects that satisfies the condition of maximizing the similarity of objects that are assigned to the same class. In addition to permitting more inferences, our study shows that the second approach results in inheritance inferences that are far more reliable, particularly in the case of exceptional subclasses.     

A simplified duality for implicative lattices and l-groups

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them.The author is supported at the Mathematical Institute of Oxford by a grant of the Argentinian Consejo de Investigations Cientificas y Tecnicas (CONICET). The author wishes to acknowledge the CONICET and the kind hospitality of the Mathematical Institute.     

Bets and Boundaries: Assigning Probabilities to Imprecisely Specified Events

Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are n-monotone, for every . When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well. Special Issue on Vagueness Edited by Rosanna Keefe and Libor Bêhounek