Aristotle’s Squares of Opposition (original) (raw)

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 ...

ARTICLE From Aristotle's Oppositions to Aristotelian Oppositions.pdf

Aristotle's philosophy is considered with respect to one central concept of his philosophy, viz. opposition. Far from being a mere side-effect of syllogistics, my claim is that opposition helps to articulate ontology and logic through what can be or cannot be in a systematic and structural way. The paper is divided into three interrelated parts. In Section 1, the notions of Being and non-Being are scrutinized through Aristotle's theory of categories. In Section 2, the notion of existence is reviewed in its ontological and logical ambiguities. In Section 3, the notion of essence is revisited in order to bring about a holist theory of meaning by individuation through opposite properties. In conclusion, the legacy of Aristotle is depicted as balanced between a powerful reflection around Being and a restrictive ontology of substance.

Some notes on the Aristotelian doctrine of opposition and the propositional calculus

Disputatio. Philosophical Research Bulletin , 2023

We develop some of Williamson's ideas regarding how propositional calculus aids in comprehending Aristotelian logic. Specifically, we enhance the utilisation of truth tables to examine the structure of opposition diagrams. Using 'conditioned truth tables', we establish logical dependency relationships between the truth values of different propositions. This approach proves effective in interpreting various texts of the Organon concerning the doctrine of opposition.

Affirmation and Denial in Aristotle’s De interpretatione

Topoi

Modern logicians have complained that Aristotelian logic lacks a distinction between predication (including negation) and assertion, and that predication, according to the Aristotelians, implies assertion. The present paper addresses the question of whether this criticism can be levelled against Aristotle's logic. Based on a careful study of the De interpretatione, the paper shows that even if Aristotle defines what he calls simple assertion in terms of predication, he does not confound predication and assertion. That is because, first, he does not understand compound assertion in terms of predication, and secondly, he acknowledges non-assertive predicative thoughts that are truth-evaluable. Therefore, the implications of Aristotle's 'predication theory of assertion' are not as devastating as the critics believe.

The Dialectical Syllogism in Aristotle's Topics

Archai , 2023

The purpose of this paper is an attempt to delimitate what the dialectical syllogism looks like in Aristotle's Topics. Aristotle never gave an example of a dialectical syllogism, but we have some clues spread over books I and VIII of the Topics which make it possible to understand at least what within a dialectical debate is a dialectical syllogism. The interpretation advanced here distinguishes the logical order of the dialectical argumentation from the order of the debate. This distinction enables us to have a better understanding of what is and how the dialectical syllogism is identified in the debate. In addition, we can solve some interpretative difficulties other interpretations could not solve, and have a more solid grasp of how endoxa are used in a dialectical debate.

Truth and Contradiction in Aristotle's De Interpretatione 6-9

Phronesis 55, pp 26-67, 2010

In De Interpretatione 6-9, Aristotle considers three logical principles: the principle of bivalence, the law of excluded middle, and the rule of contradictory pairs (according to which of any contradictory pair of statements, exactly one is true and the other false). Surprisingly, Aristotle accepts none of these without qualification. I offer a coherent interpretation of these chapters as a whole, while focusing special attention on two sorts of statements that are of particular interest to Aristotle: universal statements not made universally and future particular statements. With respect to the former, I argue that Aristotle takes them to be indeterminate and so to violate the rule of contradictory pairs. With respect to the latter, the subject of the much discussed ninth chapter, I argue that the rule of contradictory pairs, and not the principle of bivalence, is the focus of Aristotle's refutation. Nevertheless, Aristotle rejects bivalence for future particular statements.

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

Aristotle on Non-Contradiction

In M. Rossetto, M. Tsianikas, G. Couvalis and M. Palaktsoglou (Eds.) "Greek Research in Australia: Proceedings of the Eighth Biennial International Conference of Greek Studies, Flinders University June 2009". Flinders University Department of Languages - Modern Greek: Adelaide, 36-43, 2011

Aristotle’s defence of the principle of non-contradiction has been recently criticized by Graham Priest. I argue that Priest’s arguments do not work against the primary version of aristotle’s principle; Priest relies on assumptions aristotle does not, and need not, accept. However, I argue that Aristotle’s denial of the existence of points can be used to criticise his defence of non-contradiction.

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.

Truth and Bivalence in Aristotle. An Investigation into the Structure of Saying

Logik, Naturphilosophie, Dialektik. Neue internationale Beiträge zur modernen Deutung der Aristotelischen Logik (N. Öffenberger & A. Vigo, eds.), 2014

The aim of this paper is rather modest: we do not intend to reconstruct Aristotle’s theory of truth (although we are convinced that there is such a thing), and we will not try to settle the issue concerning Bivalence in Aristotle. We merely want, on the one hand, to argue for the consistency between the main Aristotelian texts on truth and a possible rejection of Bivalence; and on the other hand, to investigate the conditions of a possible counterexample to Bivalence. The motivation for this research is also very specific. We are interested in the apparent violation of Bivalence introduced by vague predicates, and in particular we want to respond to a family of arguments put forward by T. Williamson in support of the idea that allowing for exceptions to Bivalence would be incoherent. We have focused on these arguments for two reasons. On the one hand, what is allegedly threatened by a denial of Bivalence is no less than the very “nature of truth or falsity”. On the other hand, Aristotle is explicitly mentioned as one of the defendants of this “natural” conception of truth, and we are reminded about the connection between Aristotle’s theory and Tarski’s semantic conception. These arguments, therefore, give us an occasion to explore Aristotle’s analysis of the nature of truth and falsity, and to examine its connection with the Tarskian conception of truth. In particular, we would like to question the assumption, which has become a commonplace in the field of analytical philosophy, that Aristotle’s notion of truth can be encoded in the pair of disquotational biconditionals that derive from Tarski’s “T schema”.