Aristotelian and Boolean Properties of the Keynes-Johnson Octagon of Opposition

IF 0.7 1区 哲学 0 PHILOSOPHY JOURNAL OF PHILOSOPHICAL LOGIC Pub Date : 2024-06-24 DOI:10.1007/s10992-024-09765-4
Lorenz Demey, Hans Smessaert
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Abstract

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 not yet studied) from deontic logic. We show that these three KJ octagons are all Aristotelian isomorphic, but not all Boolean isomorphic to each other (the first two are representable by bitstrings of length 7, whereas the third one is representable by bitstrings of length 6). These results nicely fit within our ongoing research efforts toward setting up a systematic classification of squares, octagons, and other diagrams of opposition. Finally, obtaining a better theoretical understanding of the KJ octagons allows us to answer some open questions that have arisen in recent applications of these diagrams.

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凯恩斯-约翰逊八边形对立的亚里士多德和布尔特性
大约在 20 世纪之交,凯恩斯和约翰逊将众所周知的正方形对立扩展为八边形对立,以解释主语否定(例如 "所有非 S 都是 P")。本文的主要目的是研究凯恩斯-约翰逊(KJ)八边形对立的逻辑特性。特别是,我们将讨论 KJ 八边形的三个具体例子:主语否定的原始八边形、知识表征的当代八边形和deontic 逻辑的第三个八边形(迄今尚未研究过)。我们证明,这三个 KJ 八边形都是亚里士多德同构的,但并不都是布尔同构的(前两个八边形可用长度为 7 的位串表示,而第三个八边形可用长度为 6 的位串表示)。这些结果与我们正在进行的为正方形、八角形和其他对立图建立系统分类的研究工作不谋而合。最后,对 KJ 八边形有了更好的理论理解,我们就能回答最近在这些图的应用中出现的一些开放性问题。
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来源期刊
CiteScore
2.50
自引率
20.00%
发文量
43
期刊介绍: The Journal of Philosophical Logic aims to provide a forum for work at the crossroads of philosophy and logic, old and new, with contributions ranging from conceptual to technical.  Accordingly, the Journal invites papers in all of the traditional areas of philosophical logic, including but not limited to: various versions of modal, temporal, epistemic, and deontic logic; constructive logics; relevance and other sub-classical logics; many-valued logics; logics of conditionals; quantum logic; decision theory, inductive logic, logics of belief change, and formal epistemology; defeasible and nonmonotonic logics; formal philosophy of language; vagueness; and theories of truth and validity. In addition to publishing papers on philosophical logic in this familiar sense of the term, the Journal also invites papers on extensions of logic to new areas of application, and on the philosophical issues to which these give rise. The Journal places a special emphasis on the applications of philosophical logic in other disciplines, not only in mathematics and the natural sciences but also, for example, in computer science, artificial intelligence, cognitive science, linguistics, jurisprudence, and the social sciences, such as economics, sociology, and political science.
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