Bipolar Argumentation Frames and Contrary to Duty Obligations, Preliminary Report

In my papers [2,7], I modelled the Chisholm paradox and generally Chisholm like sequences of contrary to duty obligations by using Reactive Kripke models [4]. Reactive Kripke frames have two types of arrows: ordinary single arrows x → y indicating accessibility relations and double arrows of the form \(( u \to v) \twoheadrightarrow ( x \to y)\), indicating reactive connections. In the frames where the ordering is a tree, as it is in the models for contrary to duty obligations, the double arrow \(( u\to v) \twoheadrightarrow ( x \to y)\) can be uniquely represented by \(v \twoheadrightarrow y\). We thus get a bipolar network where we interpret → as support and \(\twoheadrightarrow \) as attack. Of course the same reactive graph can be manipulated in the Deontic way [2], when we read it as modelling contrary to duty obligations and it can be manipulated in the argumentation way [1,3], when viewed as a bipolar network. The question arises, can we find a family of tree like graphs, (which do not sacrifice generality neither in the contrary to duty area nor in the bipolar argumentation area) for which the Deontic and the argumentation manipulations are the same. This paper shows that this is possible, and thus establishes a connection between the contrary to duty area and the bipolar argumentation area. Note the following: 1 This connection with bipolar argumentation frames is made possible because of the modelling of contrary to duty obligation using reactive Kripke models. The connection between Reactivity and Bipolarity is more easy to see. 2 The way the game is played in each area is different. So we have here a wide scope for interaction and exchange of ideas between argumentation and normative reasoning. These include: 1 Deontic like modelling and axiomatisations for bipolar argumentation 2 argumentation semantics for contrary to duty paradoxes which can especially handle contrary to duty loops (a subject hardly mentioned in the contrary to duty literature). 3 The equational approach to contrary to duty, imported from the equational approach to argumentation [8] 4 The fact that bipolar frames can be instantiated as contrary to duty obligation might shed some light on the polarised debate in the argumentation community on how to instantiate argumentation networks, see [5]. 5 Settle quesions of how to model (what is) support in argumentation 6 Doing Modal Logic in Bipolar Argumentation Theory (compare with [6]).