The theory of the square of opposition has been worked out many centuries ago as a part of Aristotelian logic of terms.In spite of its inexactness (for instance it is not possible to decide whether the termsquare of opposition is a logical or a metalogical term) this theory is included without any changes in the usual elementary course of logic.The author defines the square of opposition in the language of the logic of propositions (see Def. 1.000) and derives (...) from this definition the usual laws of the square of opposition and several new theorems. (shrink)

According to Greimas, the semiotic square is far more than a heuristic for semantic and literary analysis. It represents the generative “deep structure” of human culture and cognition which “define the fundamental mode of existence of an individual or of a society, and subsequently the conditions of existence of semiotic objects” (Greimas & Rastier 1968: 48). The potential truth of this hypothesis, much less the conditions and implications of taking it seriously (as a truth claim), have received little attention (...) in the literature. In response, this paper traces the history and development of the logical square of opposition from Aristotle to Greimas and beyond, to propose that the relations modelled in these diagrams are embodied relations rooted in gestalt memories of kinesthesia and proprioception from which we derive basic structural awareness of opposition and contrast such as verticality, bilaterality, transversality, markedness and analogy. To make this argument, the paper draws on findings in the phenomenology of movement (Sheets-Johnstone 2011a, 2011b, 2012, Pelkey 2014), recent developments in the analysis of logical opposition (Beziau & Payette 2008), recent scholarship in (post)Greimasian semiotics (Corso 2014, Broden 2000) and prescient insights from Greimas himself (esp. 1968, 1984). The argument of the paper is further supported through a visual and textual content analysis of a popular music video, both to highlight relationships between the semiotic square and mundane cultural ideologies and to show how these relationships might be traced to the marked symmetries of bodily movement. In addition to illustrating the enduring relevance of Greimasean thought, the paper further illustrates the neglected relevance that embodied chiasmus holds for developments in anthropology, linguistics and the other cognitive sciences. (shrink)

. 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. (shrink)

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant 's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters have criticized Kant 's solution as artificial and prejudicial. In the paper, I do not dispute such (...) claims, but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant 's thoughts. (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.”.

In this paper I propose a set-theoretical interpretation of the logical square of opposition, in the perspective opened by generalized quantifier theory. Generalized quantifiers allow us to account for the semantics of quantificational Noun Phrases, and of other natural language expressions, in a coherent and uniform way. I suggest that in the analysis of the meaning of Noun Phrases and Determiners the square of opposition may help representing some semantic features responsible to different logical properties of these expressions. (...) I will conclude with some consideration on scope interactions between quantifiers. (shrink)

Papers... "selected from a larger number of contributions most of them based on talks presented at the First World Congress on the Square of Opposition organized in Montreux in June 2007"--Preface, p. 12.

LOGICIANS owe a considerable debt of gratitude to Ogden and Veatch for their highly satisfactory exposition of the problem of existential import in the Square of Opposition and for their stimulating efforts to solve it. The authors gave an apt description of the situation reigning in this field when they said.

We are pleased to present this special issue of the journal History and Philosophy of Logic dedicated to the square of opposition.The square of opposition is a diagram and a theory of opposition re...

The square of opposition is a diagram related to a theory of oppositions 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 quantification, 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 Opposition, 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. (shrink)

The square of opposition has never been drawn by classical Arabic logicians, such as al-Fārābī and Avicenna. However, in some later writings, we do find squares, which their authors call rather ‘tables’ (sing. _lawḥ_). These authors are Shihāb al-Dīn al-Suhrawardī and Muhammed b. Yūsuf al-Sanūsī. They do not pertain to the same geographic area, but they both provide squares of opposition. The aim of this paper is to analyse these two squares, to compare them with each other and with (...) the traditional square of opposition drawn by Apuleius and Boethius, with regard to their particular structure and their vertices and to evaluate their validity. We will show that both squares are different from each other and from the traditional square both with regard to their vertices, and to the arrangements of the oppositional relations. In addition, while al-Suhrawardī uses the notion of matter modalities to define the oppositions, al-Sanūsī relies rather on al-Rāzī’s and al-Khūnajī’s distinctions. Both authors agree, however, on the fact that affirmatives have an import while negatives do not. For this reason, their squares are valid from a logical viewpoint. As diagrams, these squares are nevertheless entirely original, due to the particular arrangements of their oppositional relations. Our analysis shows also that the later author does not seem to know the square drawn by the earlier one and that both authors do not seem to know Apuleius’ or Boethius’ squares. (shrink)

This essay explores the validity of Gregory Boyd’s open theistic account of the nature of the future. In particular, it is an investigation into whether Boyd’s logical square of opposition for future contingents provides a model of reality for free will theists that can preserve both bivalence and a classical conception of omniscience. In what follows, I argue that it can.

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)

In this paper, we introduce a Hilbert style axiomatic calculus for intutionistic logic with strong negation. This calculus is a preservative extension of intuitionistic logic, but it can express that some falsity are constructive. We show that the introduction of strong negation allows us to define a square of opposition based on quantification on possible worlds.

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is (...) represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square. (shrink)

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic. We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.

The theory of the square of opposition has been studied for over 2,000 years and has seen a resurgence in new theories and research since the second half of the twentieth century. This volume collects papers presented at the Sixth World Congress on the Square of Opposition, held in Crete in 2018, developing an interdisciplinary exploration of the theory. Chapter authors explore subjects such as Aristotle’s ontological square, logical oppositions in Avicenna’s hypothetical logic, and the power of (...) the square of opposition to solve theological problems regarding predestination and theodicy. Other topics covered include: Hegel’s opposition to diagrams De Morgan’s unpublished octagon of opposition turnstile figures of opposition institutional model-theoretic treatment of oppositions Lacan’s four formulas of sexuation the theory of oppositional poly-simplexes The Exoteric Square of Opposition will appeal to pure logicians, historians of logic, semioticians, philosophers, theologians, mathematicians, and psychoanalysts. (shrink)

Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions (...) of the square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers. (shrink)

The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.

During the last decades, the psychology of reasoning has identified experimentally many fallacies committed by spontaneous reasoners. Given these experimental results, some theories have been developed about this phenomenon, mainly algorithmic theories. This paper develops instead a computational modelling of these current fallacies which appear as simplifications in the treatment of information that do not respect the formal rules of classical propositional logic. These fallacies are explained as crushes in the Klein group structure and so, in squares of opposition. These (...) crushes are an effect of not to take into account the dual of the binary operator at work in the major premise of the inference. This analysis allows predictions on fallacies not identified before, which are fallacies produced when reasoning with incompatibilities. The paper concludes with consequences of the analysis on pedagogical strategies for the teaching of logic. (shrink)

After explaining the interdisciplinary aspect of the series of events organized around the square of opposition since 2007, we discuss papers related to the 4th World Congress on the Square of Opposition which was organized in the Vatican at the Pontifical Lateran University in 2014. We distinguish three categories of work: those dealing with the evolution and development of the theory of opposition, those using the square as a metalogical tool to give a better understanding of various (...) systems of logic and those related with applications of the theory of opposition to conceptual analysis and pedagogy. (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)

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.

It has become an article of faith among historians of logic that the square of opposition diagram is due not to Aristotle, but to Apuleius. I examine three Aristotelian texts and argue that Prior Analytics I.46 contains a square of opposition, making Aristotle the discoverer of the diagram.

. The truth conditions that Aristotle attributes to the propositions making up the traditional square of opposition have as a consequence that a particular affirmative proposition such as ‘Some A is not B’ is true if there are no Bs. Although a different convention than the modern one, this assumption remained part of centuries of work in logic that was coherent and logically fruitful.

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The (...) paper focuses mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:NECESSARY PROPOSITIONS AND THE SQUARE OF OPPOSITION MARK ROBERTS University of Rhode Island Kingston, Rhode Island IT IS COMMONPLACE to define contradictory, contrary, and subcontrary propositions in the following way: contradictory propositions cannot both be true and cannot both be false; contrary propositions cannot both be true but can both be false; and subcontrary propositions can both be true but cannot both be false. In his Introduction to (...) Logic 1 Irving Copi raises a problem with two of these definitions which he believes forces him to limit the range of propositions which can be used in the square of opposition. Since contrary propositions can both be false, but the falsity of a necessarily true proposition is not possible, a necessarily true proposition has no contrary. Therefore, only contingent propositions can be contraries. Subcontrary propositions can both be true, but the truth of a necessarily false proposition is not possible.2 This problem is rarely raised in other logic texts, but, it is interesting to note, it is mentioned by a number of neoscholastic philosophers, for example, P. Nicholas Russo,3 Rev. H. Grenier,4 1 Irving M. Copi, Introduction to Logic (New York: Macmillan, 1986), pp. 178-179. Copi first introduced this problem in the fifth edition of his work on the basis of an article by David H. Sanford entitled " Contraries and Subcontraries,'' Nous 2 (1968) : 95-96. Sanford argues that the most reasonable way to resolve the problem of necessary propositions is to assume that contingent propositions comprise the square of opposition. 2 Copi, pp. 178-179. 8 Nicholas Russo, Summa Philosophica (Boston: Apud Marlier et Socios, 1[;85), p. 23. 4 Henri Grenier, Thomistic Philosophy, Vol. I (Charlottetown, Canada: St. Dunstan's University, 1950), p. 69. 427 428 MARK ROBERTS G. Sanseverino,5 Jacques Maritain,6 F.-X. Maquart,7 P. Coffey,8 and others.9 (It is also mentioned by Bishop Whately.10 ) Each of these philosophers makes necessarily true or false propositions an exception to the definition of contrary and subcontrary propositions. The reason is they and Copi interpret the definitions of contrary and subcontrary propositions to mean that the falsity of each contrary proposition is possible and the truth of each subcontrary proposition is possible. It should be mentioned, however, that the position of the philosophers cited above is not entirely the same as Copi's. Copi reasons that these definitions imply that necessary propositions have no contrary or subcontrary. The other philosophers believe that necessary propositions can be contraries and subcontraries. Thus, in the case of necessary propositions contrary and subcontrary propositions are, instead, defined merely in terms of their quantity and quality, not by whether or not they can both be false or can both be true. In fact, Coffey and some of the other philosophers cited above assert that, when the propositions composing the square are necessary, one can infer the truth of the universal from the truth of 5 Gaietano Sanseverino, Philosophia Christiana, Vol. II (Neapoli: Vincentii Manfredi, 1862), pp. DCCXXXVIII ff. 6 J. Maritain, An Introduction to Logic (London: Sheed & Ward, 1937), p. 135, notes 1 & 2. 7 F.-X. Maquart, Elementa Philosophiae, Vol. I (Parisiis: Andreas Blot, 1937)' p. 126 ff. 8 P. Coffey, The Science of Logic, Vol. 1 (New York: Longmans, Green & Co., 1918), p. 225, 226. 9 Joseph Gredt, Elementa Philosophiae, Vol. I (Friburgi: Herder, 1929), p. 46, notes 1 & 2; Sylvester J. Hartman, A Textbook of Logic (New York: American Book Co., 1936), p. 162 ff; Roland Houde and Jerome J. Fischer, Handbook of Logic (Dubuque, Iowa: Wm. C. Brown Co., 1954), p. 61; Eduard Hugon, Cursus Philosophiae Thomisticae, Vol. I (Parisiis: P. Lethielleux n.d), p. 155; Dennis C. Kane, Logic: The Art of Predication and Inference (Providence: Providence College Press, 1978), p. 98 ff; Francis P. Siegfried, Essentialia Philosophiae (Philadelphia: Dolphin Press, 1927), p. 25; Francis Varvello, Minor Logic, trans. and supplemented Arthur D. Fearon (San Francisco: Univ. of San Francisco Press, 1933), p. 73 ff. 10 Richard Whately, Elements of Logic (Boston: James Munroe & Co., 1854), p. 77 ff. PROPOSITIONS AND THE SQUARE OF OPPOSITION 429 the particular and the falsity of the particular from the falsity of the universal.11 Maritain maintains that in the case... (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.

Jean-Yves Béziau Abstract In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and to organize world congresses on the topic with subsequent publications.

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 of these philosophers did not in fact reject empty terms, and that, when properly understood, there are no logical problems with the classical square. Instead, the classical square, compared to its modern version, raises the issue of the existential import (...) of words like all; a semantic issue. I argue that the modern square is more interesting than Parsons allows, because it presents, in contrast with the classical square, notions of negation that are ubiquitous in natural languages. This is an indirect logical argument against interpreting all with existential import. I also discuss some linguistic matters bearing on the latter issue. (shrink)

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric (...) versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme. (shrink)

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. (shrink)

To salvage traditional logic and traditional square of opposition from the problem of existential import, logicians have been offering solutions for centuries. In this paper, firstly it will be argued that as far as we know, the historically first solution proposed by Abelard in 11th century and by Seuren in 2002 is actually a version of the O-Corner Interpretation of traditional logic, which is generally attributed to the 14th-century logician Ockham. Secondly, it will be advocated that two systems of (...) Abelard and of Ockham have the same logical power. Lastly, the main claim will be that Abelard’s and Seuren’s system shall be favored over Ockham’s system. (shrink)

In this paper a set of categorical sentences called an antilogistic tetrad is presented as a perspective on Aristotle's square of opposition. An antilogistic tetrad is formed by collecting the premises of a pair of valid syllogisms the conclusions of which are contradictory categorical sentences. A set of such premises serves to bring together Aristotle's concern with debate and the syllogism, and as such may be seen as a way of “completing” Aristotle's analysis of the square of opposition.The (...) debate context is characterized by opposing views for which arguments are offered. The square of opposition captures that contending of opposing views; and is also basic to the presentation of categorical sentences, a necessary condition for the syllogism. By using C. S. Peirce's notion of abductive argument to produce the middle term, and hence to construct deductive syllogistic arguments, antilogistic tetrads may be formed on any contended subject. For that reason, the process sketched above for forming antilogistic tetrads is called “completing the square of opposition.”. (shrink)

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 may be 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. (shrink)

Can an appeal to the difference between contrary and contradictory statements, generated by a non-uniform behaviour of negation, deal adequately with paradoxical cases like the sorites or the liar? This paper offers a negative answer to the question. This is done by considering alternative ways of trying to construe and justify in a useful way (in this context) the distinction between contraries and contradictories by appealing to the behaviour of negation only. There are mainly two ways to try to do (...) so: i) by considering differences in the scope of negation, ii) by considering the possibility that negation is semantically ambiguous. Both alternatives are shown to be inapt to handle the problematic cases. In each case, it is shown that the available alternatives for motivating or grounding the distinction, in a way useful to deal with the paradoxes, are either inapplicable, or produce new versions of the paradoxes, or both. (shrink)

. This paper aims to highlight some peculiarities of the semiotic square, whose creation is due in particular to Greimas’ works. The starting point is the semiotic notion of complex term, which I regard as one of the main differences between Greimas’ square and Blanché’s hexagon. The remarks on the complex terms make room for a historical survey in Aristotle’s texts, where one can find the philosophical roots of the idea of middle term between two contraries and its (...) relation to notions such as difference, position and motion. In the Stagirite’s non-logical works, the theory of the intermediate, or middle term, represents an important link between opposition issues and ethics: this becomes a privileged perspective from which to reconsider the semiotic use of the square, i.e., its inclusion in the semio-narrative structures articulating the sense of texts. (shrink)

. In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

In the paper we build up the ontology of Leśniewski’s type for formalizing synthetic propositions. We claim that for these propositions an unconventional square of opposition holds, where a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation. Further, we construct a non-Archimedean extension of Boolean algebra and show that in this algebra just two squares of opposition are formalized: conventional and the (...) square that we invented. As a result, we can claim that there are only two basic squares of opposition. All basic constructions of the paper (the new square of opposition, the formalization of synthetic propositions within ontology of Leśniewski’s type, the non-Archimedean explanation of square of opposition) are introduced for the first time. (shrink)

Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cognitive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The (...) square is considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leaking O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute the effects of U-restriction. (shrink)

While it seems that just about everything is being excavated today, the Square of Opposition might seem a fairly unlikely candidate for such a kind of interest. The square is a very simple device o...

This special issue is related to the 6th World Congress on the Square of Opposition which took place at the Orthodox Academy of Crete in November 2018. In this introductory paper we explain the context of the event and the topics discussed.

In recent decades, research in the square of opposition has increased. New interpretations, extensions, and generalizations have been suggested, both Aristotelian and non-Aristotelian ones. The paper attempts to compare different versions of the square of opposition. For this reason, we appeal to the wider categorical model-theoretic framework of the theory of institutions.