The aim of the present paper is to analyse the archeology of the concept of contradiction, more precisely in Plato, and to reveal the influence that the latter had on Aristotle’s reflection on contradiction and contrariety. This paper will show that it is possible to find examples of a notion of contradiction in Plato’s refutative dialogues, in which Socrates is described as refuting his interlocutors by demonstrating the contrary of their initial thesis. However, Plato never used the word antiphasis (...) to name the act of contradicting oneself, but preferred the expression enantia legein heautôi, which means “to say the contrary to oneself”. (shrink)

F.H. Bradley has been characterized by many commentators as something of a sceptic. And the reason why Bradley is often cast in this light is, I believe, largely a result of his theory of relations. As even the casual student is aware, the relational nature of all judgment leads, on Bradley’s analysis, to an infinite and, many have claimed, “vicious” regress. And when we add to this Bradley’s claim that all assertion is, at some level, “contradictory”, there seems to be (...) good reason for the sceptical moniker. What I would like to suggest in this paper, though, is that Bradley’s “scepticism” is not what it seems. Hence in what follows I shall argue that Bradley’s views on contradiction and contrariety are actually important aspects of his larger theory of knowledge; and, as such, they should be understood — not as doctrines whose intent is the condemnation of our ordinary inferences — but, rather, as a crucial component in the defense of such practices. What I hope to show, then, is that, far from being an endorsement of radical scepticism, Bradley’s theory of contradiction and contrariety is an effort to preserve the continuity between one judgment and another — a continuity that he sees as necessary to any coherent account of inference. But first some preliminaries. (shrink)

Contrariety is essential to Hume's notion of experience. Its role suggests that experience itself, its specificity, and the contrariety of events must be actual, not merely hypothetical.

Each chapter heading bears a phrase from a contemporary author, held to incorporate the character of that section of the study under consideration. Chapter 1 carries the title given to early English translations of the Lettres provinciales; chapter 2 recalls the description of Pascall by Boyle and other English scientists; and chapter 3 draws from Kennett's preface to his version of the Pensees. The heading of chapter 4 is from Pope's Essay on Man. The exclamation which introduces chapter 5 concludes (...) an essay in the Gentleman's Magazine for 1789, probably by Boswell; the words for chapter 6 come from a pastoral letter of John Wesley; and chapter 7 represents the verdict of Coleridge. The title of the book itself is derived from the heading to the twenty-first chapter of Kennett's Pensees, which seems to have forecast the essence of the eighteenth-century's perplexity upon the issues raised by Pascal: 'The strange Contrarieties discoverable in Human Nature, with regard to Truth, and Happiness, and many other things.'. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:29. CONTRARIETY AND CAUSALITY IN HUME Hume's notion of contrariety ranks among the most obscure in his theory of relations. To make matters worse, the puzzling account of contrariety he offers can be shown inconsistent in the following way. The Treatise (T69-82) divides all relations into two disjoint classes - one class containing relations of knowledge (in the strict sense) ascertained by the mere comparison of (...) ideas, the other containing those of probability or 'uncertain knowledge' requiring 'experience' or empirical evidence. Causality is the paradigm of a probability relation - contrariety falls into the class of knowledge relations. Thus unlike causality the discovery of a contrary relation appears to require no experience. Yet in direct contradiction Hume claims that with the exception of existence and non-existence (which are singled out for special attention), all knowledge of contrary relations demands experience: no two ideas are in themselves contrary, except those of existence and non-existence.... All other objects, such as fire and water, heat, and cold, are only found to be contrary from experience, and from the contrariety of their 'causes ' or effects '. (T15) The charge of inconsistency appears inevitable. However, I shall argue that Hume escapes by implicitly operating with two distinct concepts of contrariety: one (which I shall term "logical contrariety") a relation of knowledge, the other (to be called "empirical contrariety") reducible to a peculiar type of causal relation, and therefore a probability relation. To adduce independent grounds for this claim I demonstrate that acceptance of these two concepts is essential if we are to explain why Hume finds 2 it important to place the following restriction on contra- 30. riety (hereafter, R): properly speaking no objects are contrary to each other, but existence an (y exists). And (4) can hardly be said to represent a form of contrariety between existent and non-existent objects. (Indeed paradox of material implication aside, (4) is an analysis of "x causes y".) Therefore to hold R and still maintain some sense of contrariety between existent and non-existent objects, Hume must restrict contrariety to logical contrariety. 32. Hume's own remarks bear out my claim. In the Treatise he recognizes that restricting contrariety to existence and non-existence means treating it as contradiction : the flattest of all contradictions... (is)... that 'tis possible for the same thing both to be and not to be. (T19, my emphasis.)... with contrariety... No one can once doubt but existence and non-existence destroy each other, and are perfectly incompatible and contrary. (T70) And in the Enquiries he interchanges the terms 'contrary' and 'negation' : The contrary of every matter of fact is still possible ; because it can never imply a contradiction... (EU25)... No negation of a fact can involve a contradiction. (EU164) This usage indicates that if? is (logically) contrary to q,? is -q, the negation of q. In this case the conjunction of? and q spells the contradiction (q & -q). The definition of logical contrariety we adopt is the same as the definition of contradiction specialized to existential sentences, so we need not state it. It is sufficient to give as a paradigm the following contradiction: (5)(Mars exists) & - (Mars exists). Here it is the existence of Mars that is logically contrary to the non-existence of Mars and therefore the contrariety restriction R is satisfied. Again it is only the existence and non-existence of the same object that is logically contrary. Empirical contrariety is most clearly defined in terms of the Lewis counterfactual G=>, where 1O' is a monadic predicate, 'e.' and 'e.' represent distinct arbitrary events, and "0(e)" means "event e occurs": 33. 5 Here it is not the existence and non-existence or occurrence (EC) e. is empirically contrary to e. iff 0(e.) D=> ~0(e.)." and non-occurrence of distinct events that gives rise to contrariety. Rather it is the existence or occurrence of one event that is empirically contrary to the existence or occurrence of another. To establish Hume's motivation for framing the condition R in such a way as to identify contrariety with contradiction, I now turn to examine the epistemological ground of the difference between empirical and logical contrariety. The ontological distinction between empirical... (shrink)

We discuss aspects of the logic of negation bearing on an issue raised by Jean-Yves Béziau, recalled in §1. Contrary- and subcontrary-forming operators are introduced in §2, which examines some of their logical behaviour, leading on naturally to a consideration in §3 of dual intuitionistic negation (as well as implication), and some further operators related to intuitionistic negation. In §4, a historical explanation is suggested as to why some of these negation-related connectives have attracted more attention than others. The remaining (...) sections (§§5, 6) briefly address a question about a certain notion of global contrariety and the provision of Kripke semantics for the various operators in play in our discussion. (shrink)

Contradiction is often confused with contrariety. We propose to disentangle contrariety from contradiction using the hexagon of opposition, providing a clear and distinct characterization of three notions: contrariety, contradiction, incompatibility. At the same time, this hexagonal structure describes and explains the relations between them.

Can an ontology which treats properties as really out there in the world be combined vvith the view that necessity is not out there? What about the necessity by which redness excludes greenness, or weighing 8 kg excludes weighing 6 kg? Armstrong, who combines property realism with logical atomism, argues that such exclusions reflect just the trivial necessity that a whole cannot be any of its proper parts. Buthis argument fails for colors themselves and for other cases of contrary properties. (...) Property realism must be necessity realism. (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them (...) from other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

The compatibilist position on the free will problem tends to be perceived as clear, rather unitary and consistent even by those who oppose it. This notion is mistaken, and is harmful to the recognition of the weaknesses and strengths of compatibilism. By examining the three main compatibilist positions and their interrelationships, I attempt to see whether compatibilists can continue to hold together the different positions; and if they cannot, which position they should remain with. The conclusions reached are that compatibilists (...) ought to opt for one (‘control’) type of compatibilism, but that compatibilism is only partially convincing. (shrink)

The compatibilist position on the free will problem tends to be perceived as clear, rather unitary and consistent even by those who oppose it. This notion is mistaken, and is harmful to the recognition of the weaknesses and strengths of compatibilism. By examining the three main compatibilist positions and their interrelationships, I attempt to see whether compatibilists can continue to hold together the different positions; and if they cannot, which position they should remain with. The conclusions reached are that compatibilists (...) ought to opt for one (‘control’) type of compatibilism, but that compatibilism is only partially convincing. (shrink)

Although it is not younger than other areas of non-classical logic, paraconsistent logic has received full recognition only in recent years, largely due to the work of, among others, Newton da Costa, Graham Priest, Diderik Batens, and Jerzy Perzanowski. A logical system Λ is paraconsistent if there is a set of Λ-formulas Δ ∪ {A} such that in Λ one may derive from Δ both A and its negation, and the deductive closure of Δ with respect to Λ is different (...) from the set of all formulas. If from Δ one may derive a formula and its negation, Δ is said to be syntactically inconsistent. But is every syntactically inconsistent set of formulas contradictory? In classical logic and many non-classical logics, every syntactically inconsistent set is unsatisfiable, that is, semantically inconsistent. If contradictoriness means semantical inconsistency, there is, up to logical equivalence, only one contradiction. In paraconsistent logics, there are usually many non-equivalent formulas representing the semantically unique contradiction in a non-paraconsistent logic. So when is a formula the contradiction of another formula, and, moreover, how does the notion of contradiction relate to the notions of contrariety and negation? (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Hume’s account of contrariety in Book I of the Treatise poses several interpretive puzzles. I consider each in turn and offer a novel interpretation of contrariety based on Hume’s discussion of the passions. That Book II and Book I form a complete chain of reasoning suggests that the way in which passions are related is analogous to the way in which ideas are related in the understanding. I argue that Hume identifies three species of empirical contrariety in (...) Book II: contraries, oppositions, and contradictions. All three species help clarify Hume’s remarks concerning contrariety in Book I. I further identify a second genus I call logical contrariety by drawing on Hume’s discussion of fictitious duration and distance. (shrink)

Four known three-valued logics are formulated axiomatically and several completeness theorems with respect to nonstandard intuitive semantics, connected with the notions of information, contrariety and subcontrariety is given.

Anton views the Aristotelian contraries as "principles of understanding, generic concepts, employed in the analysis of any determinate process whatever." He argues that the principle of contrariety simply renders process intelligible and is not, as it was for many of Aristotle's predecessors, a causal principle. In the course of his argument the author shows the use of this "formal demand for determinateness" in widely diverse areas, proceeding from the categories to ontology and language, and through psychology to ethics.--R. P.

In developing his theory of aether in De Caelo 1, Aristotle argues, in DC 1.4, that one circular motion cannot be contrary to another. In this paper, I discuss how Aristotle can maintain this position and accept the existence of celestial spheres that rotate in contrary directions, as he does in his revision of the Eudoxan theory in Metaphysics 12.8.

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another (...) type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean (...) terms and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

This paper offers a new interpretation of John Austin’s views both on assertion and on adverbs, as result of which an expressivist thesis concerning the semantics for action sentences is advanced. First, Austin’s analysis of assertion based on various, specific assertive forces and his remarks on adverbs are systematically connected in order to obtain assertive schemata for action sentences. Finally, those schemata are put to work as the expression of inferential commitments implicit in argumentative practices of different sorts (exculpatory, justificatory (...) and illustrative) in the deployment of which, both logical contrariety and contradiction are exploited. (shrink)

This article embodies the result of a collective work that seeks to understand the manner in which the notion of conflict is transformed by the attempt to apply it to reality, according to the doctrines hereby comprise as “ontologies of conflict”. It seems that there are three resulting “logics of conflict” which may receive the following qualifications: logic of contradiction, logic of contrariety and logic of difference.

The examples Hume relies on in _Treatise_ I.iv.2 raise questions about the role of contrariety in experience as it affects belief in the objects of perception.

Some legal theorists say that legal entrapment to commit a crime is incoherent. So far, there is no satisfactorily precise statement of this objection in the literature: it is obscure even as to the type of incoherence that is purportedly involved. (Perhaps consequently, substantial assessment of the objection is also absent.) We aim to provide a new statement of the objection that is more precise and more rigorous than its predecessors. We argue that the best form of the objection asserts (...) that, in attempting to entrap, law-enforcement agents lapse into a form of practical incoherence that involves the attempt simultaneously to pursue contrary ends. We then argue that the objection, in this form, encompasses all cases of legal entrapment only if it is supplemented by appeal to the premise that law-enforcement agents have an absolute duty never to create crimes. (shrink)

the theory of modal groups is an abstract theory that may be used to show formal similarities between various formulations of epistemic logics (eg Hintikka, Chisholm, and von Wright) and formal systems in other domains.

The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name (...) problem and it is not clear what is the intuitive notion corresponding to it. We explain then that the triangle of contrariety proposed by different people such as Vasiliev and Jespersen solves these problems, but that we don’t need to reject the square. It can be reconstructed from this triangle of contrariety, by considering a dual triangle of subcontrariety. This is the main idea of Blanché’s hexagon. We then give different examples of hexagons to show how this framework can be useful to conceptual analysis in many different fields such as economy, music, semiotics, identity theory, philosophy, metalogic and the metatheory of the hexagon itself. We finish by discussing the abstract structure of the hexagon and by showing how we can swing from sense to non-sense thinking with the hexagon. (shrink)