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). (shrink)

In 2006, mountaineer David Sharp died on the slopes of Mount Everest. Sharp’s death led to public outrage after allegedly 40 climbers passed by the dying Sharp on their way to the peak, without stopping to help. But, since the slopes of Everest are a high-risk environment and rescuing Sharp would have entailed great risks for the rescuers, it is not clear whether the other mountaineers had a moral duty to rescue him. In a recent article, Patrick Findler introduces a (...) principle to analyse such cases which states that we have a duty to rescue under dangerous circumstances, if the involved risks are not higher than the risks we are already taking in the pursuit of our own, morally less worthy ends. However, Findler then rejects this principle as too demanding. In this paper I will defend the principle against its inventor and argue in favour of such a duty. And while it may be true that the other mountaineers were justified in passing Sharp for different reasons, the principle shows that not only climbers, but adventure-sport athletes in general and people who engage in high-risk endeavours, may have a duty to rescue that doesn’t apply to others. (shrink)

The article expands the traditional system of concepts used in deontic logic, in order to allow the inclusion of supererogatory behaviour. This requires the development of a deontic decagon. In addition, it is shown how this decagon can be used to interpret deontic terms, e.g. in Islamic Law.