Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings (...) to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

This book traces the development of formal logic from its origins inancient Greece to the present day. The authors first discuss the work oflogicians from Aristotle to Frege, showing how they were influenced by thephilosophical or mathematical ideas of their time. They then examinedevelopments in the present century.

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, (...) where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure. (shrink)

We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness and to (...) account for a bigger typological puzzle, namely the absence of lexicalizations of ‘XOR’ (= exclusive or), ‘all-or-none’, and many other imaginable logical operators, one must restrict the basic lexicalizable elements to a small set of primitives. We suggest that an ordering based perspective, following Keenan and Faltz (Boolean semantics for natural language, 1985), makes the stipulated primitives that we arrive at more natural. We also propose a modification to Horn’s proposal, based on recent work on implicatures, in which only the implicature condition is operative and in which markedness is part of the definition of the alternatives for scalar implicatures rather than an independent condition. (shrink)

This book offers a unique synthesis of past and current work on the structure, meaning, and use of negation and negative expressions, a topic that has engaged thinkers from Aristotle and the Buddha to Freud and Chomsky. Horn's masterful study melds a review of scholarship in philosophy, psychology, and linguistics with original research, providing a full picture of negation in natural language and thought; this new edition adds a comprehensive preface and bibliography, surveying research since the book's original publication.

This paper compares two 3D logical diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron and the nested tetrahedron. Geometric properties such as collinearity and central symmetry are examined from a cognitive perspective, focussing on diagram design principles such as congruence/isomorphism and apprehension.

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then (...) the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. (shrink)

The traditional Christian view that God foreknows the future exclusively in terms of what will and will not come to pass is partially rooted in two ancient Hellenistic philosophical assumptions. Hellenistic philosophers universally assumed that propositions asserting ‘ x will occur’ contradict propositions asserting ‘ x will not occur’ and generally assumed that the gods lose significant providential advantage if they know the future partly as a domain of possibilities rather than exclusively in terms of what will and will not (...) occur. Both assumptions continue to influence people in the direction of the traditional understanding of God's knowledge of the future. In this essay I argue that the first assumption is unnecessary and the second largely misguided. (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

The paper proceeds in four parts. First, a theory of liberty is sketched, which theory depends heavily upon three distinctions in ethics. These distinctions are shown to be maintainable using no more than resources of a standard deontic logic. That logic is itself recast from the trial of the required, the forbidden, and the optional into a simpler logic of the obligatory and the permitted. Secondly, two challenges presented to this theory of liberty and its use of standard deontic logic, (...) are examined. These challenges stem from the seeming existence of actions and omissions that are neither forbidden or required, on the one hand, nor morally indifferent, on the other. Such actions are subclassified as supererogatory , suberogatory , or quasi-erogatory . Third, attempts to reduce these three kinds of not indifferent but not obligatory actions into either indifference of a kind, or obligation of a kind, are rejected. Fourth, the moral permissibility of these three kinds of not indifferent but not obligatory actions is defended, and the non-punishability of such permissible if non-otimific actions is defended on retributivist grounds. Der Artikel besteht aus vier Teilen. Im ersten Teil wird eine Freiheitstheorie skizziert, die auf drei Differenzierungen in der Ethik basiert. Allein auf Grundlage der Standard-deontischen Logik wird gezeigt, daß die Differenzierungen aufrechterhalten werden können. Die Standard-Logik selbst, gebildet aus der Trias von "geboten", "verboten" und "erlaubt", wird in eine einfachere Logik des Obligatorischen und des Erlaubten umgeformt. Im zweiten Teil werden zwei Kritikpunkte an der Freiheitstheorie und ihrem Gebrauch der üblichen deontischen Logik untersucht. Die Kritikpunkte ergeben sich aus der Tatsache, daß es offenbar Handlungen und Unterlassungen gibt, die einerseits weder verboten noch geboten, andererseits aber auch nicht moralisch indifferent sind. Solche Handlungen werden eingeteilt in die supererogatorischen , die suberogatorischen und die quasi-erogatorischen Handlungen. Als Drittes werden verschiedene Versuche verworfen, die drei Typen nicht indifferenter, aber auch nicht obligatorischer Handlungen entweder auf eine wie auch immer geartete Indifferenz oder auf eine wie auch immer geartete Obligation zu reduzieren. Schließlich wird die These vertreten, daß die drei Typen nicht indifferenter und nicht obligatorischer Handlungen moralisch erlaubt sind, und daß erlaubte, wenn auch nicht optimale Handlungen - auf der Grundlage einer Theorie der Vergeltung - nicht strafbar sind. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles ("contraposition" (...) and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

© Springer International Publishing Switzerland 2016. We develop a systematic approach for dealing with informationally equivalent Aristotelian diagrams, based on the interaction between the logical properties of the visualized information and the geometrical properties of the concrete polygon/polyhedron. To illustrate the account’s fruitfulness, we apply it to all Aristotelian families of 4-formula fragments that are closed under negation and to all Aristotelian families of 6-formula fragments that are closed under negation.

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical (...) statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. (shrink)

Contemporary philosophers sometimes present the complex relationships that can exist between permission, precept and prohibition within a given structure of law in a language of symbolic logic or in illustrative diagrams. Other modern scholars have pointed out that early formulations of the basic ideas they employ can be found in writers of the seventeenth and eighteenth centuries, including Leibnitz and Bentham and, especially, the German jurist Gottfried Achenwall. This article shows that the same structure of ideas was included centuries earlier (...) in the work of Marsilius of Padua, and that his argument too can be presented as a modern 'deontic hexagon'. It concludes with a discussion of how Marsilius' engagement with the contemporary dispute over Franciscan poverty helped to shape his teaching in this area. (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

1. Aristotle’s doctrine of the mean—hereafter DM—still provokes discussion. To my knowledge this discussion has not yet singled out an important logical pattern latent in DM. I think this pattern can help explain Aristotle’s failure to “… speak in terms of rules of conduct which apply equally to all men, and which all can understand.”.

_William of Sherwood's Introduction to Logic _ was first published in 1966. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The _Introduction to Logic _ by William of Sherwood, of which this is the first English translation, is the oldest surviving treatise which contains a treatment of the most distinctive and interesting medieval contributions to logic and semantics. Sherwood was a master at Oxford (...) and Paris in the thirteenth century and the author of several logical treatises. Besides presenting material of interest in its own right, this volume is useful as an introduction to the study of those aspects of medieval philosophy that are most pertinent to the interests of contemporary philosophers. Professor Kretzmann has provided biographical, bibliographical, and philosophical backgrounds on Sherwood and an analytical table of contents. (shrink)

This paper contains two traditions of diagrammatic studies namely one, the Euler–Venn–Peirce diagram and the other, following tradition of Aristotle, the square of oppositions. We put together both the traditions to study representations of singular propositions, their negations and the inter relationship between the two. Along with classical negation we have incorporated negation of another kind viz. absence. We have also considered the changes that take place in the context of open universe.

Burgess-Jackson has recently suggested that the debate between theism and atheism can be represented by means of a classical square of opposition. However, in light of the important role that the position of agnosticism plays in Burgess-Jackson’s analysis, it is quite surprising that this position is not represented in the proposed square of opposition. I therefore argue that the square of opposition should be extended to a slightly larger, more complex Aristotelian diagram, viz., a hexagon of opposition. Since this hexagon (...) does represent the position of agnosticism, it arguably yields a more helpful representation of the theism/atheism debate. It would be naïve to presume that Aristotelian diagrams can, by themselves, lead to a comprehensive solution of debates as intricate as that between theism and atheism. Nevertheless, this paper aims to show that these diagrams — especially if they are chosen carefully — have an important methodological role to play, by systematically organizing and clarifying the debate. (shrink)

© The Author 2018. Published by Oxford University Press. All rights reserved. Logical geometry provides a broad framework for systematically studying the logical properties of Aristotelian diagrams. The main aim of this paper is to present and illustrate the foundations of a computational approach to logical geometry. In particular, after briefly discussing some key notions from logical geometry, I describe a logical problem concerning Aristotelian diagrams that is of considerable theoretical importance, viz. the task of finding the maximal Boolean complexity (...) of a given family of Aristotelian diagrams, and I then present and discuss a simple algorithm for automatically solving this task. This algorithm is naturally implemented within the paradigm of logic programming. In order to illustrate the theoretical fruitfulness of this algorithm, I also show how it sheds new light on several well-known families of Aristotelian diagrams. (shrink)

A simple division of all possible states into Rechtsstaaten and Unrechtsstaaten may be insufficient. Hence a more refined conceptual system should be considered. Kant distinguishes four concepts: republicanism, despotism, barbarism and anarchism . Using four rather than two concepts permits one to classify republics as Rechtsstaaten and barbaric states as Unrechtsstaaten , while despotic states fall under a third independent category. Despotic and barbaric states, however, can now be classified under the new concept Un-Rechtsstaaten , where despotic states are designated (...) as "with law" and barbaric states as "without law". Anarchism, then is neither a Rechtsstaat nor an Un-Rechtsstaat . This article examines Kant's conceptual system and considers whether it is complete. Sofern eine Einteilung denkbarer Staatstypen in einerseits Rechtsstaaten und andererseits Unrechtsstaaten nicht ausreichend erscheint, muß über eine Erweiterung der Begriffssystematik nachgedacht werden. Hilfreich ist hier ein Begriffssystem, das Kant vorschlägt, indem er drei Staatstypen unterscheidet: Republik, Despotie und Barbarei, wobei noch als Abgrenzungsbegriff die Anarchie hinzutritt. In seiner Schrift Anthropologie in pragmatischer Hinsicht unterscheidet Kant diese vier Strukturen mit Hilfe der Parameter Gesetz, Freiheit und Gewalt wie folgt: A. Gesetz und Freiheit ohne Gewalt . B. Gesetz und Gewalt ohne Freiheit . C. Gewalt ohne Freiheit und Gesetz . D. Gewalt mit Freiheit und Gesetz . Man kann in dieser Systematik die Republik mit dem Rechtsstaat und die Barbarei mit dem Unrechtsstaat identifizieren; die Despotie verbleibt als dritter unabhängiger Staatstyp. Despotie und Barbarei fallen jetzt allerdings gemeinsam unter den neu zu bildenden Begriff des Un-Rechtsstaats. Analog zur Begriffsbildung bei Kant unterscheiden sich Despotie und Barbarei dadurch, daß erstere durch den Parameter "mit Gesetz", letztere durch den "ohne Gesetz" ausgezeichnet ist. Der vorliegende Beitrag befaßt sich u.a. damit, wie die Passage in der Anthropologie im einzelnen gedeutet werden kann, und schließlich mit der Frage, ob sich sinnvoll noch weitere Staatstypen von den bereits genannten unterscheiden lassen. Dazu wird auf eine Reihe paralleler Überlegungen Kants in den Reflexionen zur Anthropologie einzugehen sein. (shrink)