NDLI: The Geometry of Standard Deontic Logic

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The Geometry of Standard Deontic Logic

Content Provider	Springer Nature Link
Author	Moretti, Alessio
Copyright Year	2009
Abstract	Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).
Starting Page	19
Ending Page	57
Page Count	39
File Format	PDF
ISSN	16618297
Journal	Logica Universalis
Volume Number	3
Issue Number	1
e-ISSN	16618300
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2009-05-26
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	logical square logical hexagon logical bi-simplexes modal logic deontic logic opposition theory oppositional geometry modal graphs Modal logic Philosophical and critical Classical propositional logic Subsystems of classical logic Partition relations Logic Computer Science Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Logic

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The Power of the Hexagon

The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

Deontological Square, Hexagon, and Decagon: A Deontic Framework for Supererogation

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