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.     

Interpolation and Amalgamation; Pushing the Limits. Part I

Continuing work initiated by Jónsson, Daigneault, Pigozzi and others; Maksimova proved that a normal modal logic (with a single unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property (cf. [Mak 91], [Mak 79]). The aim of this paper is to extend the latter result to a large class of logics. We will prove that the characterization can be extended to all algebraizable logics containing Boolean fragment and having a certain kind of local deduction property. We also extend this characterization of the interpolation property to arbitrary logics under the condition that their algebraic counterparts are discriminator varieties. We also extend Maksimova's result to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than 2, too.The problem of extending the above characterization result to no n-normal non-unary modal logics remains open.Related issues of universal algebra and of algebraic logic are discussed, too. In particular we investigate the possibility of extending the characterization of interpolability to arbitrary algebraizable logics.     

Classifying ℵo-Categorical Theories II: The Existence of Finitely Axiomatizable Proper Class II Theories

Clark and Krauss [1977] presents a classification of complete, satisfiable and _o-categorical theories in first order languages with finite non-logical vocabularies. In 1988 the first author modified this classification and raised three questions about the distribution of finitely axiomatizable theories. This paper answers two of those questions.     

A Way to Interpret Łukasiewicz Logic and Basic Logic

Fuzzy logics are in most cases based on an ad-hoc decision about the interpretation of the conjunction. If they are useful or not can typically be found out only by testing them with example data. Why we should use a specific fuzzy logic can in general not be made plausible. Since the difficulties arise from the use of additional, unmotivated structure with which the set of truth values is endowed, the only way to base fuzzy logics on firm ground is the development of alternative semantics to all of whose components we can associate a meaning. In this paper, we present one possible approach to justify ex post Łukasiewicz Logic as well as Basic Logic. The notion of ambiguity is central. Our framework consists of a Boolean or a Heyting algebra, respectively, endowed with an equivalence relation expressing ambiguity. The quotient set bears naturally the structure of an MV- or a BL-algebra, respectively, and thus can be used to interpret propositions of the mentioned logics.     

The enduring scandal of deduction

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.     

思维聚焦对发展型文化消费产品购买意愿的影响

本研究探析了消费者的思维聚焦（过程聚焦/结果聚焦）对发展型文化消费产品购买意愿的影响,发现结果聚焦的消费者比过程聚焦的消费者对发展型文化消费产品的购买意愿更强烈。同时,时间距离对这一效应存在调节作用,即时间距离远时,这一效应显著;时间距离近时,思维聚焦对发展型文化消费产品的购买意愿没有显著影响。此外,本研究进一步考察了时间距离较远时,消费者的未来时间洞察力部分中介了思维聚焦对发展型文化消费产品的购买意愿。     

Toward An Explanatory Model of Innovative Behavior

Research on innovative behavior has persistently focused on determining the correlations between numerous demographic and psychographic variables and specific external actions. Unfortunately, such research has consistently produced unreliable results. A preferred alternative to this line of research is an analysis of the causes of innovative behavior with the objective of understanding the process underlying this activity. A framework is proposed for examining innovative behavior based upon a comprehensive model of variety seeking, of which innovative behavior is one outcome. The framework attempts to depict the causes of, the processes involved, and the outcomes of variety seeking, particularly as it relates to innovative behavior. A number of propositions are presented.     

Non-monotonic Probability Theory and Photon Polarization

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov’s positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov’s theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov’s other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov’s theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works.     

The long-term effect of relational information in classification learning

This study examines the long-term effect of mutual information in the learning of Shepardian classifications. Mutual information is a measure of the complexity of the relationship between features because it quantifies how the features relate to each other. For instance, in various categorisation models, Type VI concepts—originally studied by Shepard, Hovland, and Jenkins (1961)—are unanimously judged to be the most complex kind of 3-D Boolean concepts. This has been largely confirmed by empirical data. Yet, it is apparently inconsistent with the fact that this concept entails the greatest amount of mutual information of all the 3-D Boolean concepts. The present study was aimed at verifying whether individuals can use relational information, in the long run, to devise easier strategies for category learning. Subject performance was measured repeatedly for 1 hour on either successive Type VI concepts (using different features between problems) or successive Type IV concepts. The results showed that shortly after the second problem, Type VI concepts became easier to learn than Type IV ones. The gap between the mean per-problem error rates of the two concepts continued to increase as the number of problems increased. Two other experiments tended to confirm this trend. The discussion brings up the idea of combining different metrics in categorisation models in order to include every possible way for subjects to simplify the categorisation process.     

The Scope and Limitations of Algebras: Some Historical and Philosophical Considerations

An important feature of mathematics, both pure and applied, during the nineteenth century was the widening from its common form to a proliferation, where the “objects” studied were not numbers or geometrical magnitudes but operations such as functions and differentiation and integration, abstract ones (as we now call them), linear algebras of vectors, matrices and determinants, and algebras in logic. In this article the author considers several of them, including the contributions of Hermann Grassmann and Benjamin Peirce. A notable feature of these developments was analogising from one algebra to another by adopting some of the same laws, such as associativity, commutativity and distributivity. In the final section we consider the normally secular character of these algebras.