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Logic-Sensitivity and bitstring semantics in the Square of Opposition

JOURNAL OF PHILOSOPHICAL LOGIC 2023年第6期52卷 1703-1721页

作者： Demey, Lorenz Frijters, Stef Katholieke Univ Leuven Ctr Log & Philosophy Sci Leuven Belgium Katholieke Univ Leuven Inst Artificial Intelligence Leuven Belgium

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the precise details of their interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term 'bitstring semantics'!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective.

关键词： Square of opposition Logical geometry bitstring semantics Logic-sensitivity Combinatorial bitstrings Semantic bitstrings

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Aristotelian and Boolean Properties of the Keynes-Johnson Octagon of Opposition

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JOURNAL OF PHILOSOPHICAL LOGIC 2024年第5期53卷 1265-1290页

作者： Demey, Lorenz Smessaert, Hans Katholieke Univ Leuven Ctr Log & Philosophy Sci Leuven Belgium Katholieke Univ Leuven Inst Artificial Intelligence Leuven Belgium Katholieke Univ Leuven Res Grp Formal & Computat Linguist Leuven Belgium

Around the turn of the 20th century, Keynes and Johnson extended the well-known square of opposition to an octagon of opposition, in order to account for subject negation (e.g., statements like 'all non-S are P'). The main goal of this paper is to study the logical properties of the Keynes-Johnson (KJ) octagons of opposition. In particular, we will discuss three concrete examples of KJ octagons: the original one for subject-negation, a contemporary one from knowledge representation, and a third one (hitherto not yet studied) from deontic logic. We show that these three KJ octagons are all Aristotelian isomorphic, but not all Boolean isomorphic to each other (the first two are representable by bitstrings of length 7, whereas the third one is representable by bitstrings of length 6). These results nicely fit within our ongoing research efforts toward setting up a systematic classification of squares, octagons, and other diagrams of opposition. Finally, obtaining a better theoretical understanding of the KJ octagons allows us to answer some open questions that have arisen in recent applications of these diagrams.

关键词： Square of opposition Octagon of opposition Aristotelian diagram J. N. Keynes W. E. Johnson bitstring semantics Logical geometry

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On the Logical Geometry of Geometric Angles

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LOGICA UNIVERSALIS 2022年第4期16卷 581-601页

作者： Smessaert, Hans Demey, Lorenz Katholieke Univ Leuven Dept Linguist Leuven Belgium Katholieke Univ Leuven Ctr Log & Philosophy Sci Leuven Belgium

In this paper we provide an analysis of the logical relations within the conceptual or lexical field of angles in 2D geometry. The basic tripartition into acute/right/obtuse angles is extended in two steps: first zero and straight angles are added, and secondly reflex and full angles are added, in both cases extending the logical space of angles. Within the framework of logical geometry, the resulting partitions of these logical spaces yield bitstring semantics of increasing complexity. These bitstring analyses allow a straightforward account of the Aristotelian relations between angular concepts. In addition, also relational concepts such as complementary and supplementary angles receive a natural bitstring analysis.

关键词： Aristotelian relations bitstring semantics Logical geometry Acute right obtuse angle Zero straight angle Reflex full angle Complementary supplementary angles

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A Diagrammatic Representation of Hegel's Science of Logic 12th

A Diagrammatic Representation of Hegel's <i>Science of Logic...

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12th International Conference on the Theory and Application of Diagrams (Diagrams)

作者： Pluder, Valentin Lemanski, Jens Univ Siegen Philosoph Seminar Siegen Germany Fernuniv Inst Philosoph Hagen Germany

ISBN: (纸本)9783030860622;9783030860615

In this paper, we interpret a 19th century diagram, which is meant to visualise G.W.F. Hegel's entire method of the Science of Logic on the basis of bitwise operations. For the interpretation of the diagram we use a binary numeral system, and discuss whether the anti-Hegelian argument associated with it is valid or not. The reinterpretation is intended to make more precise rules of construction, a stricter binary code and a review of strengths and weaknesses of the critique.

关键词： Diagrammmatic representation GWF Hegel bitstring semantics Boolean algebra Logic Contradictions Dialectic

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Schopenhauer's Partition Diagrams and Logical Geometry 12th

Schopenhauer's Partition Diagrams and Logical Geometry

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12th International Conference on the Theory and Application of Diagrams (Diagrams)

作者： Lemanski, Jens Demey, Lorenz Fernuniv Inst Philosophy Hagen Germany Katholieke Univ Leuven Ctr Log & Philosophy Sci Leuven Belgium

ISBN: (纸本)9783030860622;9783030860615

The paper examines Schopenhauer's complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer's partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) a-structures.

关键词： Arthur Schopenhauer Logic diagrams Partition diagram Aristotelian diagram bitstring semantics Logical geometry

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Computing the maximal Boolean complexity of families of Aristotelian diagrams

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JOURNAL OF LOGIC AND COMPUTATION 2018年第6期28卷 1323-1339页

作者： Demey, Lorenz Katholieke Univ Leuven Ctr Log & Philosophy Sci Leuven Belgium

Logical geometry provides a broad framework for systematically studying the logical (and other) 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 particular, Prolog). 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.

关键词： Aristotelian diagram logical geometry bitstring semantics Aristotelian family Boolean subfamily logic programming

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