The Power of the Hexagon (original) (raw)
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Logica Universalis, 2012
The square of opposition and many other geometrical logical figures have increasingly proven to be applicable to different fields of knowledge. This paper seeks to show how Blanché generalizes the classical theory of oppositions of propositions and extends it to the structure of opposition of concepts. Furthermore, it considers how Blanché restructures the Apuleian square by transforming it into a hexagon. After presenting G. Kalinowski's formalization of Blanché's hexagonal theory, an illustration of its applicability to mathematics, to modal logic, and to the logic of norms is depicted. The paper concludes by criticizing Blanché's claim according to which, his logical hexagon can be considered as the objective basis of the structure of the organisation of concepts, and as the formal structure of thought in general. It is maintained that within the frame of diagrammatic reasoning Blanché's hexagon keeps its privileged place as a "nice" and useful tool, but not necessarily as a norm of thought.
new-dimenions-of-the-square-of-opposition-philosophia.pdf
The square of opposition is a diagram related to a theory of opposi-tions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantifi-cation, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has recently become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Oppo-sition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers.
Logica Universalis, 2012
The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the "logical square" (or "square of opposition"), of which it is a much more general and powerful "relative". The discovery of the former did not raise interest, neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, "sentenced to death" by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with "oppositions") has appeared, "oppositional geometry" (also called "n-opposition theory", "NOT"), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of "logical bi-simplexes of dimension m", itself just one term of the more general infinite series (of series) of the "logical poly-simplexes of dimension m". In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of "hybrid logical hexagon", we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change.
In On Interpretation, Chapter 7, Aristotle alters a system of three pairs of natural mutually contradictory propositions, in that he eliminates the pair where two natural universals Men are white and Men are not white oppose each other contradictorily (see the diagram I above). This alteration has serious consequences : the two natural pairs, which Aristotle considers exclusively: All men are white versus Some men are not white and Some men are white versus No man is white are illegitimately identified with the two pairs of logical contradictories constituting the logical square: A versus O and I versus E respectively
The metalogical hexagon of opposition
The difference between truth and logical truth is a fundamental distinction of modern logic promoted by Wittgenstein. We show here how this distinction leads to a metalogical triangle of contrariety which can be naturally extended into a metalogical hexagon of oppositions, representing in a direct and simple way the articulation of the six positions of a proposition vis-à-vis a theory. A particular case of this hexagon is a metalogical hexagon of propositions which can be interpreted in a modal way. We end by a semiotic hexagon emphasizing the value of true symbols, in particular the logic hexagon itself.
The Geometry of Logical Opposition
The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.
New light on the square of oppositions and its nameless corner
2003
It has been pointed out that there is no primitive name in natural and formal languages for one corner of the famous square of oppositions. We have all, some and no, but no primitive name for not all. It is true also in the modal version of the square, we have necessary, possible and impossible, but no primitive name for not necessary.
On the Historical Transformations of the Square of Opposition as Semiotic Object
Logica Universalis, 2020
In this paper, we would show how the logical object "square of opposition", viewed as semiotic object (articulated in textual or/and diagrammatic code), has been historically transformed since its appearance in Aristotle's texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle's original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle's works is transformed into a diagrammatic one, based on a new "reading" of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval "Neo-Aristotelian" reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the "square" of opposition into a "triangle", where the points standing for the O and I forms are contracted into one point, the M(I, O) form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new "Non-Aristotelian" logic (and associated ontology), i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in (Neo-)Aristotelian style and the context of discovery is foundational (analogical to Lobachevsky's research on the axiomatics of geometry). It establishes a bifurcation (proliferation) point in the development of logic. No unique logic is recognized, but different logics concerning different domains (ontologies, respectively). One branch
The Square of Opposition and the Four Fundamental Choices
Logica Universalis, 2008
Each predicate of the Aristotelian square of opposition includes the word "is". Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of the square of opposition is achieved by not adjoining, according to two Leibniz' suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek's culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded.