Traditional Square Of Opposition Research Articles

A Cube of Opposition for Predicate Logic

The traditional square of opposition is generalized and extended to a cube of opposition covering and conveniently visualizing inter-sentential oppositions in relational syllogistic logic with the usual syllogistic logic sentences obtained as special cases. The cube comes about by considering Frege–Russell’s quantifier predicate logic with one relation comprising categorical syllogistic sentence forms. The relationships to Buridan’s octagon, to Aristotelian modal logic, and to Klein’s 4-group are discussed. The photo shows a prototype sculpture for the cube.

O-Köşesi Yorumu Nedir ve Geleneksel Karşıtlık Karesini Kurtarabilir mi?

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.

Система реєстрації прав Торренса: її «принципи» і переважаючий принцип справедливості (Torrens System: Its 'Principles' and the Overriding Principle of Fairness)

Система реєстрації прав Торренса: її «принципи» і переважаючий принцип справедливості (Torrens System: Its 'Principles' and the Overriding Principle of Fairness)

Retorica e logica nella critica di Lorenzo Valla del quadrato delle opposizioni

In this paper I discuss Lorenzo Valla's criticism of the traditional Square of Opposition displayed in the second book of the Dialectica. I show that, according to Valla, the opposition rules of the propositions must take into account both common speeches and the correct use of Latin language, not the formal link occurring between the parts of propositions. In Valla's perspective, this theoretical change is carried out through rhetoric and philology, and it involves a reassessment of the arts of the trivium. As regards this topic, I argue that Valla aims niether to reduce dialectic to rhetoric nor to replace the former with the latter, but rather to establish some rhetorical principles as a better-suited way to set the opposition rules, because they take into account the linguistic context in which these rules apply.

Logical Squares for Classical Logic Sentences

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 (CPC). We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

Prior on Aristotle’s Logical Squares

This paper introduces Prior’s unpublished paper Aristotle on Logical Squares, which is deposited in the Bodleian Library and which discusses Greniewski’s definition of the $$\Box $$ operator, which Greniewski introduced in his paper Proba ‘odmlodzenia’ kwadratu logicznego. It is a unique attempt to formalize the square of opposition. Bendiek’s review, which is an important intermediary between Greniewski’s and Prior’s paper, is also mentioned here. Greniewski’s main motivation was to rejuvenate the traditional square of opposition in order to make a square of opposition more precise. He did not investigate the history of the square of opposition. Prior’s approach differs a great deal from Greniewski’s one because he bases his analyses of Greniewski’s operator on Aristotle’s definitions of squares of oppositions. There are some queries, which arise due to the different aims of Greniewski and Prior and their different approach might have caused Prior’s effort to the approximation of Greniewski’s operator to Aristotle’s definitions of the squares of opposition was unsuccessful.

Aristotle and Łukasiewicz on Existential Import

ABSTRACT:Jan Łukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing the contemporary understanding of Aristotle's logical systems. However, Łukasiewicz's interpretation is based on a number of tendentious claims, not least, on the claim that the syllogistic was intended to apply only to nonempty terms. I show that this interpretation is not true to Aristotle's text and that a more coherent and faithful interpretation admits empty terms while maintaining all the relations of the traditional square of opposition.

Kant on Existential Import

AbstractThis article reconstructs Kant's view on the existential import of categorical sentences. Kant is widely taken to have held that affirmative sentences (theAandIsentences of the traditional square of opposition) have existential import, whereas negative sentences (EandO) lack existential import. The article challenges this standard interpretation. It is argued that Kant ascribes existential import only to some affirmative synthetic sentences. However, the reasons for this do not fall within the remit of Kant's formal logic. Unlike traditional logic and modern standard quantification theory, Kant's formal logic is free from existential commitments.

The Medieval Octagon of Opposition for Sentences with Quantified Predicates

The traditional Square of Opposition consists of four sentence types. Two are universal and two particular; two are affirmative and two negative. Examples, where ‘S’ and ‘P’ designate the subject and the predicate, are: ‘every S is P’, ‘no S is P’, ‘some S is P’ and ‘some S is not P’. Taking the usual sentences of the square of opposition, quantifying over their predicates exhibits non-standard sentence forms. These sentences may be combined into non-standard Squares of Opposition (an Octagon in this case), and they reveal a new relationship not found in the usual Square. Medieval logicians termed ‘disparatae’ pairs of sentences like ‘every S is some P’ and ‘some S is every P’, which are neither subaltern nor contrary, neither contradictory nor subcontrary. Walter Redmond has designed a special language L to express the logical form of these sentences in a precise way. I will use this language to show how Squares of Opposition, standard and non-standard, form a complex network of relations which bring to light the subtleties contained in this traditional doctrine.

El octágono medieval de Oposición para oraciones con predicados cuantificados

El cuadrado tradicional de oposición consta de cuatro clases de oraciones: dos universales y dos particulares, dos afirmativas y dos negativas. Ejemplos, donde “S” y “P” designan sujeto y predicado, son: “todo S es P”, “Ningún S es P”, “Algún S es P” y “Algún S no es P”. Tomando estas oraciones y cuantificando sobre los predicados obtenemos formas no usuales que pueden ser combinadas en cuadrados no usuales de oposición (un octágono en este caso) y que muestran una relación que no está en el cuadrado tradicional. Los lógicos medievales llamaron disparatae a oraciones como “Todo S es algún P” y “Algún S es todo P”. Walter Redmond ha diseñado un lenguaje especial L para expresar, de manera precisa, la forma lógica de estas oraciones. Usaré este lenguaje para mostrar cómo los cuadrados de oposición usual e inusual forman una compleja red de relaciones que muestran la complejidad de esta doctrina tradicional.

Constraints on the lexicalization of logical operators

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.

From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory

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 Blanche’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.

El octagon medieval de oposición y equivalencia: tres aplicaciones / The Medieval Octagon of Opposition and Equivalence: Three Applications

I describe an octagon of opposition and equivalence developed by fourteenth-century logicians, in particular by Jean Buridan in his Summulae de dialectica. This «square» of opposition displays complex logical relations, one of which is not found in the traditional square of opposition. The octagon allows expression of three kinds of sentences: quantified modal sentences, oblique sentences, and sentences with quantified predicates. The octagon shows that medieval logicians were working with a logic of relations, an identity logic, and a modal logic not unlike the logic of our own day.

Anselm’s Logic of Agency

The origins of treating agency as a modal concept go back at least to the 11th century when Anselm, Archbishop of Canterbury, provided a modal explication of the Latin facere ‘to do’, which can be formalized within the context of modern modal logic and neighborhood semantics. The agentive logic induced by his conception satisfies the traditional square of opposition, but also has some unique properties which reflect the fact that Anselm’s modal view of agency is grounded strongly in non-logical philosophical and theological considerations. We show that the logic modeling Anselm’s theory of agency provides an interesting alternative to standard logics of agency based on stit-theory.

Things That are Right with the Traditional Square of Opposition

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.

Ploucquet’s “Refutation” of the Traditional Square of Opposition

In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”.

Weakened Semantics and the Traditional Square of Opposition

In this paper we present a proposal that (i) could validate more relations in the square than those allowed by classical logic (ii) without a modification of canonical notation neither of current symbolization of categorical statements though (iii) with a different but reliable semantics.

The existential assumptions of traditional logic

There have been and continue to be disagreements about how to consider the traditional square of opposition and the traditional inferences of obversion, conversion, contraposition and inversion from the perspective of contemporary quantificational logic. Philosophers have made many different attempts to save traditional inferences that are invalid when they involve empty classes. I survey some of these attempts and argue that the only satisfactory way of saving all the traditional inferences is to make the existential assumption that both the subject and predicate classes and their complement classes are non-empty for all the propositions we admit. I briefly indicate the room for continued controversy over how properly to interpret Aristotle’s statements regarding these inferences, but find some plausibility in the views of Manley Thompson and A.N.Prior that Aristotle had in mind a particular arrangement of existential import unfamiliar to most contemporary logicians

Aristotelian Logic (review)

130 JOURNAL OF THE HISTORY OF PHILOSOPHY 32:1 JANUARY 1994 definition, "prudent fortitude" (phronimoskart~,6a). Its downfall is caused by the identification of "prudence" with technical skill. (A more detailed argument in favor of "prudence " as a translation would have been helpful.) Schmid claims that Laches conceives of "prudence in a merely technical sense~ and that he cannot "distinguish prudence from art" (123). However, it is Socrates who, in examining what phronimos means, offers examples borrowed from the technai.In other words, it is he who suggests techni as a possible model for the prudence that is courage. Since he eventually rejects his own suggestion, it is at least possible that Plato's intention here is to point to a conception of moral knowledge that is precisely not a techn/. For Schmid, the main lesson of the refutation is that it reflects "an inherent conflict in the traditional Greek ideal of honor and virtue" (123). Perhaps it does, but it is nonetheless possible that some version of the Bonitz thesis can yet be maintained; perhaps the refutation is meant to suggest a "nontechnical" conception of moral knowledge . If so, then the dialogue is simultaneously aporetic and productive. It ends without an explicitly successful definition of courage, but supplies the materials from which such a definition can be constructed. This view, which I think is plausible, varies significandy with Schmid's basic thesis. Nevertheless, Schmid's own argument is powerful and must be taken seriously. This is a fine book, rich in detail and nuance. It is filled with probing observations and clearheaded arguments. On Manly Courageshould be consulted by every serious student of the Laches. Its prose is lucid and it is written with conviction. Schmid takes Plato seriously. For him the dialogue is more than an antiquarian's delight or a convenient opportunity to exercise his academic skills. Instead, the Laches is a book that can teach its readers important lessons about courage. This reader was left with the impression that Schmid has long cared and wondered about the idea of courage, and that he has turned to Plato for guidance. His commentary is designed to help others do the same. DAVID ROOCHNIK WilliamsCollege William T. Parry and Edward A. Hacker. AristotelianLogic.Albany: State University of New York Press, 1991. Pp. x + 545. Cloth, $49-5o- Paper, $16.95. Readers of this journal may find this book interesting and useful for several reasons. First, it treats the topics of traditional or Aristotelian logic--terms, definitions, standard categorical propositions, the traditional square of opposition, immediate inference , categorical syllogisms, sorites, enthymemes, hypothetical and disjunctive syllogisms , fallacies--as they have been treated in the history of logic and philosophy. Most notably, the authors devote a whole chapter to presenting Aristotle's deductive system for reducing all valid forms of categorical syllogism to certain primitive syllogistic forms, taken as axiomatic. The authors thus present far more than just the concepts of traditional logic and instructions on how to apply them. They are concerned with the theory behind these notions and want to develop that theory accurately in accord with traditional approaches. In the course of doing this, they present copious historical BOOK REVIEWS 131 references, both in footnotes and appendices, which should constitute an additional source of interest to historians of philosophy. Indeed, the books ends with a minihistory of logic from Aristode through G6del, with mention of major contributors in the ancient, medieval, and modern periods. This book is intended as a college text. I have serious reservations about this. The book's style is extremely dry and academic. I often wondered whether students would have any interest in the concepts defined and illustrated (I fincl little motivation to pique that interest), or with how much enthusiasm they would master the rules and definitions presented, sometimes with daunting multiplicity. It would be unfortunate if a student perceived a major component of his or her logic course as simply a body of rules and definitions to be memorized and then promptly forgotten. My impression is that for beginning logic students the text is written at too high a level, demanding a commitment to the subject and a level of sophistication and familiarity...

Truth-Functional Conditionals and Modern vs. Traditional Syllogistic

The failure of nine traditionally valid categorical syllogisms in modern logic,2 as well as the failure of implications, contrariety and sub-contrariety among A, E, I, and 0 propositions with identical subject and predicate terms, is customarily explained by references to the modern doctrine of 'existential import'. But this is a mistake. We wish to show that the differences between what should be counted as laws of logic are due, instead, to the employment of two different conditionals, one truth-functional and one not, in the two logics. By traditional logic we shall mean the standard account given in modern textbooks of the theory of the syllogism as developed from Aristotle's time down through the Port Royale Logic, with 24 valid categorical syllogisms out of 256 possible, and the theory of immediate inference as expressed in the traditional square of opposition, including conversion of E and I, contraposition of A and 0 and obversion of A, E, I, and 0 propositions. By modern logic we mean the logic of Frege, Russell, and Whitehead through the first-order predicate calculus. For convenience we shall use Quine's formulation in giving proofs later in the paper.3