Square of Opposition: A Diagram and a Theory in Historical Perspective (original) (raw)

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.

Around and Beyond the Square of Opposition

The theory of oppositions based on Aristotelian foundations of logic has been pictured in a striking square diagram which can be understood and applied in many different ways having repercussions in various fields: epistemology, linguistics, mathematics, sociology, physics. The square can also be generalized in other two-dimensional or multi-dimensional objects extending in breadth and depth the original Aristotelian theory. The square of opposition from its origin in antiquity to the present day continues to exert a profound impact on the development of deductive logic. Since 10 years there is a new growing interest for the square due to recent discoveries and challenging interpretations. This book presents a collection of previously unpublished papers by high level specialists on the square from all over the world.

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.

FALLACY OF THE SQUARE OF OPPOSITION

The heart of Aristotelian Logic is the square of opposition. This study engaged on further [re]investigation and meta-logical analysis of the validity of the square of opposition. Further, in this paper, it has been modestly established, with greater clarity, the exposition of the strengths, more than the presentation of the defects, loopholes and weaknesses, of the Aristotelian Logic in a descriptive and speculative manner. The unconcealment of the breakdown of the square of opposition marked a rupture and the opening of avenues of alternative reasoning. The critical and analytical exposition of the loopholes of the square of opposition led to a realization that things around us could have been and still be different; and there could have been better alternative reasoning than what we have called, adopted, and worshipped [Greek] logic. Results show that the downfall of the oppositional relationships in the square of opposition provided a proof of the logical illusion of Aristotle or the loophole of Traditional Logic. The laws of opposition, that have been considered the measures of logically deductive inferences, are practically almost totally logical deceptions. By implication, if the laws of subcontrariety, contrariety, and subalternation [and maybe contradiction] have collapsed, the square of opposition has also collapsed; hence, Aristotle’s square of opposition is a fallacy. This means that the square of opposition has errors and in itself an error.

On the Aristotelian square of opposition

Kapten Mnemos Kolumbarium, en festskrift med …, 2005

A common misunderstanding is that there is something logically amiss with the classical square of opposition, and that the problem is related to Aristotle's and medieval philosophers' rejection of empty terms. But [Parsons 2004] convincingly shows that most ...

Aristotle’s Squares of Opposition

2018

The article argues that Aristotle’s Square of Opposition is introduced within a context in which there are other squares of opposition. My claim is that all of them are interesting and related to the traditional Square of Opposition. The paper focuses on explaining this textual situation and its philosophical meaning. Apart from the traditional Square of Opposition, there are three squares of opposition that are interesting to follow: the square of opposition with privative terms (19b19-24), the one with indefinite-term oppositions (20a20-23), and the modal square (22a24-31), which are all contained in Aristotle’s De Interpretatione 10 and 13. The paper explains that all these squares follow a common plan, which is to demonstrate that every affirmation has its own negation, whatever is the proposition either categorical or conditional, or modal or non-modal, which is a reference to the universal importance of contradiction in logic.

The Origin of the Opposition

2019

The opposition πτώσι" όρθη (ευθεία) / πτώσει" πλάγιοι, which with time began to express the contrast between the nominative and the oblique cases (casus rectus casus obliqui) in the grammatical tradition, first appeared in the Greek reflection on language most probably in the circle of the Stoic doctrine, where it was used to determine the meanings of nouns per­ ceived from the point of view of their constituting elements of the predicative-argumentative structures which formed propositions (αξιώματα). What justifies this statement is the fact that in the framework of the Stoic dialectics concepts denoted by terms όρθη πτώσι" and πλάγιοι πτώσει" were unambiguously situated in the sphere of the linguistically expressed content (τα σημαινόμενα, τα λεκτά) and used consistently in connection with the concept of κατηγόρημα (‘predicate’), that is the predicative content expressed by the verb. The analysis of the preserved records demonstrates that the term όρθη πτώσι&q...

The Square of Opposition: A Cornerstone of Thought

Studies in Universal Logic, 2017

We first describe how after having started in Montreux, Switzerland in 2007, the congress on the square of opposition moved to the American University of Beirut in Lebanon in 2012 after a stop at the University Pasquale Paoli in Corsica in 2010. We then describe the square congress at the Pontifical Lateran University in the Vatican in 2014 and the resulting publications.