Free Łukasiewicz and Hoop Residuation Algebras

Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which is a copy of the {, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops).     

The semantic side of decision making

The research reported in this paper follows the perspective that decision making is a meaningful act that conveys information. Furthermore, the potential meanings associated with decision options may affect the decisions themselves. This idea is examined in the contexts of compensation, donation, and exchange. In general, judgments were relation dependent and meaning dependent. Furthermore, the results show nonmonotonicities and limited substitutability in a pattern that challenges straightforward ways of mapping decisions onto a common currency of utility.     

Induction as conditional probability judgment

Existing research on category-based induction has primarily focused on reasoning about blank properties, or predicates that are designed to elicit little prior knowledge. Here, we address reasoning about nonblank properties. We introduce a model of conditional probability that assumes that the conclusion prior probability is revised to the extent warranted by the evidence in the premise. The degree of revision is a function of the relevance of the premise category to the conclusion and the informativeness of the premise statement. An algebraic formulation with no free parameters accurately predicted conditional probabilities for single- and two-premise conditionals (Experiments 1 and 3), as well as problems involving negative evidence (Experiment 2).     

Out of sorts? Some remedies for theories of object concepts: a reply to Rhemtulla and Xu (2007)

Responds to comments made by Rhemtulla and Xu on the current authors' original paper Concepts of individual objects (e.g., a favorite chair or pet) include knowledge that allows people to identify these objects, sometimes after long stretches of time. In an earlier article, the authors set out experimental findings and mathematical modeling to support the view that judgments of identity depend on people's beliefs about the causal connections that unite an object's earlier and later stages. This article examines M. Rhemtulla and F. Xu's (2007) critique of the causal theory. The authors argue that M. Rhemtulla and F. Xu's alternative sortal proposal is not a necessary part of identity judgments, is internally inconsistent, leads to conflicts with current theories of categories, and encounters problems explaining empirical dissociations. Previous evidence also suggests that causal factors dominate spatio-temporal continuity and perceptual similarity in direct tests. The authors conclude that the causal theory provides the only existing account consistent with current evidence. (PsycINFO Database Record (c) 2007 APA, all rights reserved).     

Understanding political responsibility in corporate citizenship: towards a shared responsibility for the common good

In this article, we explore the debate on corporate citizenship and the role of business in global governance. In the debate on political corporate social responsibility it is assumed that under globalization business is taking up a greater political role. Apart from economic responsibilities firms assume political responsibilities taking up traditional governmental tasks such as regulation of business and provision of public goods. We contrast this with a subsidiarity-based approach to governance, in which firms are seen as intermediate actors who have political co-responsibilities in society endowed upon them by (inter)national governmental institutions. We argue that both approaches face conceptual and empirical problems, and do not make clear the content and scope of political corporate responsibility. Based on Iris Marion Young’s account of political responsibility we argue that corporate actors and governmental actors have a shared responsibility to tackle societal problems. Taking political corporate responsibility not only entails engaging in private action or engaging in public–private partnerships, but it also includes aiding governmental actors to remedy injustice or even create public institutions where they do not yet exist. By adding this perspective we contribute to the debate on responsibility in corporate citizenship and clarify the political role business can play in global governance.     

Equivalence of Consequence Operations

This paper is based on Lectures 1, 2 and 4 in the series of ten lectures titled “Algebraic Structures for Logic” that Professor Blok and I presented at the Twenty Third Holiday Mathematics Symposium held at New Mexico State University in Las Cruces, New Mexico, January 8-12, 1999. These three lectures presented a new approach to the algebraization of deductive systems, and after the symposium we made plans to publish a joint paper, to be written by Blok, further developing these ideas. That project was still incomplete when Blok died. In fact, there is no indication that he had prepared a draft of the paper, and we do not know what new material he intended to include. I am therefore not in a position to complete the project as he had envisioned it. So, I have settled for the more limited objective of presenting the material from the three lectures, leaving to others the task of adapting the techniques used there to more general situations. Preface by Bjarni Jónsson Dedicated to the memory of Willem Johannes Blok     

Protoalgebraic logics

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension _U of is finitely axiomatized (provided has only finitely many inference rules); (II) _U has only finitely many extensions.