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Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic—not only did she study under C. S. Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which logical problem did Ladd-Franklin solve in her thesis, and (...) |
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The objects called cubes of opposition have been presented in the literature in discordant ways. The aim of the paper is to offer a survey of such various kinds of cubes and evaluate their relation with an object, here called “Aristotelian cube”, which consists of two Aristotelian squares and four squares which are semiaristotelian, i.e. are such that their vertices are linked by some so-called Aristotelian relation. Two paradigm cases of Aristotelian squares are provided by propositions written in the language (...) |
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The first part of the paper aims at showing that the notion of an Aristotelian square may be seen as a special case of a variety of different more general notions: the one of a subAristotelian square, the one of a semiAristotelian square, the one of an Aristotelian cube, which is a construction made up of six semiAristotelian squares, two of which are Aristotelian. Furthermore, if the standard Aristotelian square is seen as a special ordered 4-tuple of formulas, there are (...) |
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The aim of this work is to revisit the proposal made by Dag Westerståhl a decade ago when he provided a modern reading of the traditional square of opposition and of related structures. We propose a formalization of this modern view and contrast it with the classical one. We discuss what may be a modern hexagon of opposition and a modern cube, and show their interest in particular for relating quantitative expressions. |
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This paper first investigates logical characterizations of different structures of opposition that extend the square of opposition in a way or in another. Blanché’s hexagon of opposition is based on three disjoint sets. There are at least two meaningful cubes of opposition, proposed respectively by two of the authors and by Moretti, and pioneered by philosophers such as J. N. Keynes, W. E. Johnson, for the former, and H. Reichenbach for the latter. These cubes exhibit four and six squares of (...) |
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Around the turn of the 20th century, Keynes and Johnson extended the well-known square of opposition to an octagon of opposition, in order to account for subject negation (e.g., statements like ‘all non-S are P’). The main goal of this paper is to study the logical properties of the Keynes-Johnson (KJ) octagons of opposition. In particular, we will discuss three concrete examples of KJ octagons: the original one for subject-negation, a contemporary one from knowledge representation, and a third one (hitherto (...) |
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With this paper I aim to demonstrate that a look beyond the Aristotelian square of opposition, and a related non-conservative view on logical determiners, contributes to both the understanding of Aristotelian syllogistics as well as to the study of quantificational structures in natural language. |
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Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →. |
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