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. (shrink)

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. (shrink)

Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → p or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number m is 2 ℵ 0 . Furthermore, we show that (...) there exists an immediate predecessor of classical logic (axiomatized by $p \leftrightarrow \square p$ ) which is not characterized by any finite algebra. The existence of modal logics having 2 ℵ 0 immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra. (shrink)

We study modal logics in the setting of varieties of modal algebras. Any variety of modal algebras generated by a finite algebra — such, a variety is called tabular — has only finitely many subvarieties, i.e. is of finite height. The converse does not hold in general. It is shown that the converse does hold in the lattice of varieties of K4-algebras. Hence the lower part of this lattice consists of tabular varieties only. We proceed to show that there is (...) a continuum of pretabular varieties of K4-algebras — those are the non-tabular varieties all of whose proper subvarieties are tabular — in contrast with Maksimova's result that there are only five pretabular varieties of S4-algebras. (shrink)

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

The present paper is a study in abstract algebraic logic. We investigate the correspondence between the metalogical Beth property and the algebraic property of surjectivity of epimorphisms. It will be shown that this correspondence holds for the large class of equivalential logics. We apply our characterization theorem to relevance logics and many-valued logics.

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such a matrix, (...) wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

Let S denote the variety of Sugihara algebras. We prove that the lattice (K) of subquasivarieties of a given quasivariety K S is finite if and only if K is generated by a finite set of finite algebras. This settles a conjecture by Tokarz [6]. We also show that the lattice (S) is not modular.

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociski, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations. (shrink)

The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van Benthem [2] the frame (...) was an essential tool to find simple examples of incomplete logics, axiomatized by a formula in two proposition letters of degree 2, or by a formula in one proposition letter of degree 4 (the degree of a modal formula is the maximal number of nested occurrences of the necessity operator in ). In [3] we showed that the modal logic determined by the veiled recession frame is incomplete, and besides that, is an immediate predecessor of classical logic (or, more precisely, the modal logic axiomatized by the formula pp), and hence is a logic, maximal among the incomplete ones. Considering the importance of the modal logic determined by the veiled recession frame, it seems worthwhile to ask for an axiomatization, and in particular, to answer the question if it is finitely axiomatizable. In the present paper we find a finite axiomatization of the logic, and in fact, a rather simple one consisting of formulas in at most two proposition letters and of degree at most three. (shrink)

In this essay we discuss Tarski's work on what he calledthe methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert,metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbookIntroduction to logic[41m].1Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions (...) of the discipline under consideration must be defined in terms of a small group of primitive expressions that seem immediately understandable. Furthermore, only those statements of the discipline are accepted as valid that can be deduced by precisely defined and universally accepted means from a small set of axioms whose validity seems evident. The method of constructing a discipline in strict accordance with these principles is known as thedeductive method, and the disciplines constructed in this manner are calleddeductive systems. Since contemporary mathematical logic is one of those disciplines that are subject to these principles, it itself is a deductive science. Tarski then goes on to say:“The view has become more and more common that the deductive method is the only essential feature by means of which the mathematical disciplines can be distinguished from all other sciences; not only is every mathematical discipline a deductive theory, but also, conversely, every deductive theory is a mathematical discipline”.This identification of mathematics with the deductive sciences is in our view one of the distinctive aspects of Tarski's work. Another characteristic feature is his broad view of what constitutes the domain of metamathematical investigations. A clue to this aspect of his work can also be found in Chapter 6 ofIntroduction to logic. After a discussion of the notions of completeness and consistency, he remarks that the investigations concerning these topics were among the most important factors contributing to a considerable extension of the domain of methodological studies, and caused even a fundamental change in the whole character of the methodology of deductive sciences. (shrink)

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). (shrink)

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

We present our personal view on W.J. Blok's contribution to modal logic.

Art history on the edge : iconology, media, and visual culture -- Four fundamental concepts of image science -- Image science -- Image X text -- Realism and the digital image -- Migrating images : totemism, fetishism, idolatry -- The future of the image : Rancière's road not taken -- World pictures : globalization and visual culture -- Media aesthetics -- There are no visual media -- Back to the drawing board : architecture, sculpture, and the digital image -- Foundational (...) sites and occupied spaces -- Border wars : translation and convergence in politics and media -- Art X environment -- The historical uncanny : phantoms, doubles, and repetition in the war on terror -- The spectacle today : a response to retort -- Coda : for a sweet science of images. (shrink)

This article considers how people judge the identity of objects (e.g., how people decide that a description of an object at one time, t₀, belongs to the same object as a description of it at another time, t₁). The authors propose a causal continuer model for these judgments, based on an earlier theory by Nozick (1981). According to this model, the 2 descriptions belong to the same object if (a) the object at t₁ is among those that are causally close (...) enough to be genuine continuers of the original and (b) it is the closest of these close-enough contenders. A quantitative version of the model makes accurate predictions about judgments of which a pair of objects is identical to an original (Experiments 1 and 2). The model makes correct qualitative predictions about identity across radical disassembly (Experiment 1) as well as more ordinary transformations (Experiments 2 and 3). (shrink)

A new English translation of Schelling’s unfinished magnum opus, complete with a contextualizing introduction by the translator.

Historically, scientific and engineering expertise has been key in shaping research and innovation policies, with benefits presumed to accrue to society more broadly over time. But there is persistent and growing concern about whether and how ethical and societal values are integrated into R&I policies and governance, as we confront public disbelief in science and political suspicion toward evidence-based policy-making. Erosion of such a social contract with science limits the ability of democratic societies to deal with challenges presented by new, (...) disruptive technologies, such as synthetic biology, nanotechnology, genetic engineering, automation and robotics, and artificial intelligence. Many policy efforts have emerged in response to such concerns, one prominent example being Europe's Eighth Framework Programme, Horizon 2020, whose focus on “Responsible Research and Innovation” provides a case study for the translation of such normative perspectives into concrete policy action and implementation. Our analysis of this H2020 RRI approach suggests a lack of consistent integration of elements such as ethics, open access, open innovation, and public engagement. On the basis of our evaluation, we suggest possible pathways for strengthening efforts to deliver R&I policies that deepen mutually beneficial science and society relationships. (shrink)

Part novella, part philosophy, Clara was Schelling's most popular work during his lifetime, and appears here in English for the first time.

Schelling's 1806 polemic against Fichte, and his last major work on the philosophy of nature. The heat of anger can concentrate the mind. Convinced that he had been betrayed by his former collaborator and colleague, Schelling attempts in this polemic to reach a final reckoning with Fichte. Employing the format of a book review, Schelling directs withering scorn at three of Fichte’s recent publications, at one point likening them to the hell, purgatory, and would-be paradise of Fichtean philosophy. The central (...) bone of contention is the understanding of nature: Fichte sees it as lifeless matter in motion, sheer opposition to be overcome, while Schelling waxes poetic in his defense of a living, organic nature of which humanity is a vital part. Indeed, we do not know ourselves without understanding our connection to nature, argues Schelling, anticipating many thinkers in contemporary environmental ethics. Dale E. Snow’s introduction sets the stage and explains the larger context of the conflict, which was already visible in the correspondence of the two philosophers, which broke off by 1802. Notes are included throughout the text, providing background information and identifying the many references to Fichte. Dale E. Snow is Associate Professor of Philosophy at Loyola University of Maryland and the author of Schelling and the End of Idealism, also published by SUNY Press. (shrink)

"This book contains ideas to develop interdisciplinary and transdisciplinary views on economy and society. It aims to disseminate heterodox ideas on various subjects related to economics and global society. The book is organised in six parts. Part 1 contains the key lectures of Backhaus on the concept of state sciences and of Klamer on the importance of culture for economics. Parts 2- 6 contain successively contributions in the areas of economic paradigms and theories, population and society, corporate issues, environment, and (...) international relations. Examples of the content are: - the changes of family life cycles due to the rise of non-traditional households; - subjective and objective inflation rates after the introduction of the Euro; - the economics of genetic engineering; - the contribution of foreign direct investment to the economic development of host countries; - the inaccuracy of economic models applied in places characterized by extreme income disparities; - the improvement of political and corporate governance; - evolutionary thinking and emission trading; - freedom and order in the European Union; - the reform of social policy in Europe. The book provides interesting creative multi-disciplinary ideas with respect to various topical issues concerning the contemporary global society. It is highly recommended for economists and social scientists in search of broad views on social science and society.". (shrink)

Through clear explanation of its characteristic concepts and doctrines, and paying close attention to the published works of its philosophers, the volume ...

When a single beam-splitter receives two beams of bosons described by Fock states (Bose-Einstein condensates at very low temperatures), interesting generalizations of the two-photon Hong-Ou-Mandel effect take place for larger number of particles. The distributions of particles at two detectors behind the beam splitter can be understood as resulting from the combination of two effects, the spontaneous phase appearing during quantum measurement, and the quantum angle. The latter introduces quantum “population oscillations”, which can be seen as a generalized Hong-Ou-Mandel effect, (...) although they do not always correspond to even-odd oscillations. (shrink)

W. J. Mander presents the first ever synoptic history of British Idealism, the school of thought which dominated English-language philosophy from the 1860s to the early 20th century. He restores to its proper place this neglected period of philosophy, introducing the exponents of Idealism and explaining its distinctive concepts and doctrines.

Although agricultural mechanization, hybrid cultivars, the Green Revolution, and modern biotechnology have been implemented on a global scale, conventional farmers still have to face a lot of ethical debate and criticism. From a plants ethics perspective, they are challenged by organic farmers to reorganize their practices in a responsible way, and no longer consider plants as mere instruments. However, by embracing the development of precision farming, conventional crop production seems to turn in the opposite direction and tends to become large-scale (...) digitalized robot farming. By integrating many kinds of data, precision farming technologies can be conceived as creating ‘digital’ plants as next step in the instrumentalization of crop plants. In this chapter we first discuss the main approaches to plant ethics. We use the concept of “flourishing” as our starting point for addressing the moral status of plants. On this basis, we develop a framework for ‘digital’ plants and apply it to several key technologies that have influenced the development of precision farming: sensing technologies, data processing and utilization, and control technologies. Finally, we discuss the possibility to develop a responsible precision farming of crop plants, based on a plant ethics of flourishing. (shrink)

The practice of killing day-old chicks in the Dutch egg sector is a recurrent subject of societal debate. Preventing the killing of young animals and in ovo sex determination are the two main alternatives for this problem available. An online questionnaire was held to ask the opinion of the Dutch public about these alternatives. The results show that no alternative will be fully accepted, or accepted by more than half of Dutch society. However, the survey does provide an insight to (...) what people think is important for their choice: food safety and a good treatment of animals. Irrespective of the alternative chosen, these values should be safeguarded and communicated clearly. (shrink)

In 1980, Jacques Derrida published La carte postale: De Socrate a Freud et au-del a. At the occasion of the 30th anniversary of the English translation, Going Postcard: The Letter(s) of Jacques Derrida revisits this seminal work in Derrida's oeuvre. Derrida himself described The Post Card in his preface as "the remainders of a destroyed correspondence," stretching from 1977 to 1979. A cryptic text, it is riddled with gaps, word plays, and a meandering analysis of the interface between philosophy and (...) psychoanalysis. The contributors who offered the fourteen essays gathered in Going Postcard were each provided with a deceptively simple task: to write a gloss to a fragment from the first part of The Post Card, "Envois." The result is a prismatic array of commentaries, excursions, and interpretations that take Derrida "to the letter." The different glosses on lemmas such as genre, erasure, telepathy, philately, and sperm transport The Post Card into the twenty-first century and offer a "correspondence," if fragmentary, with Derrida's work and the work to come. (shrink)

What precisely, W. J. T. Mitchell asks, are pictures (and theories of pictures) doing now, in the late twentieth century, when the power of the visual is said to be greater than ever before, and the "pictorial turn" supplants the "linguistic turn" in the study of culture? This book by one of America's leading theorists of visual representation offers a rich account of the interplay between the visible and the readable across culture, from literature to visual art to the mass (...) media. (shrink)

"[Mitchell] undertakes to explore the nature of images by comparing them with words, or, more precisely, by looking at them from the viewpoint of verbal language.... The most lucid exposition of the subject I have ever read."—Rudolf Arnheim, _Times Literary Supplement_.

W. J. Mander provides a brief introduction to and critical assessment of the thought of the greatest of the British Idealist philosophers, F. H. Bradley (1846-1924), whose work has been largely neglected in this century. After a general introduction to Bradley's metaphysics and its logical foundations, Mander shows that much of Bradley's philosophy has been seriously misunderstood. Mander argues that any adequate treatment of Bradley's thought must take full account of his unique dual inheritance from the traditions of British empiricism (...) and Hegelian rationalism. The scholarship of recent years is assessed, and new interpretations are offered of Bradley's views about truth, predication, and relations, and of his arguments for idealism. (shrink)