“McNamara’s ‘deontic dodecagon for DWE’ in the light of Oppositional Geometry”

Abstract

Oppositional geometry, i.e. the study of the “oppositional figures”, has long been approached in a rather random way. In deontic logic, beyond the classical “deontic square” (a particular application of the logical square, or square of opposition), this has given birth in 1972 to Kalinowski’s “deontic hexagon” (a particular application of Jacoby’s, Sesmat’s and Blanché’s “logical hexagon”), to Joerden’s “deontic decagon” (1987), to McNamara’s “deontic dodecagon” and “deontic octodecagon” (1996) and to Wessels’ “deontic decagon” and “deontic hexadecagon” (2002, 2004). Now, since 2004 there is a formal, mathematically founded theory of all these kinds of structures, a new flourishing branch of logic and geometry, “N-Opposition Theory” (for short: “NOT”), also called “oppositional geometry”. This general theory of the oppositions among n terms shows that after the logical square (n=2) and hexagon (n=3), there is a logical cube (n=4), and that these three oppositional solids belong to an infinite series of “oppositional bi-simplexes of dimension m” (in fact, the theory tells much more). Using NOT, in this paper we examine McNamara’s “deontic dodecagon”, which aims at expressing this author’s system DWE (for “Doing Well Enough”), one of the standard models for dealing logically with “supererogation”. After showing that, despite the fact that its underlying DWE system is logically sound and complete (as proven by Mares and McNamara in 1997), the oppositional geometry presented as being a “deontic dodecagon” is mistaken (for in NOT’s terms this polygon is irremediably both oppositionally redundant and oppositionally incomplete) we show how to correct it, strongly but successfully, within the NOT framework.