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.

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of (...) import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

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)

This volume examines the entire logical and philosophical production of Nikolai A. Vasil’ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet. The author frames Vasil’ev’s work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning of (...) the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers. (shrink)

This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports...

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)

Jan Lukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing contemporary understanding of Aristotle's logical systems. However, Lukasiewicz's interpretation is based on a number of tendentious claims, not least, the claim that the syllogistic was intended to apply only to non-empty 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.

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 main objective of this paper is devoted to two main parts. First, the paper introduces logical interpretations of classical structures of opposition that are constructed as extensions of the square of opposition. Blanché’s hexagon as well as two cubes of opposition proposed by Morreti and pairs Keynes–Johnson will be introduced. The second part of this paper is dedicated to a graded extension of the Aristotle’s square and Peterson’s square of opposition with intermediate quantifiers. These quantifiers are linguistic expressions such (...) as “most”, “many”, “a few”, and “almost all”, and they correspond to what are often called “fuzzy quantifiers” in the literature. The graded Peterson’s cube of opposition, which describes properties between two graded squares, will be discussed at the end of this paper. (shrink)