Normative systems represented by Boolean quasi-orderings (Q2718313)

Normative systems are, for example, the body of law of a country, or portions of the law. This paper proposes to represent normative systems by means of Boolean quasi-orderings, structures \(\langle B,\wedge,',R\rangle\), where \(\langle B,\wedge,'\rangle\) is a Boolean algebra with \(\top\) and \(\bot\) as top and bottom elements, and \(R\) is a quasi-ordering on \(B\) that satisfies the conditions (i) if \(aRb\) and \(aRc\) then \(aR(b\wedge c)\), (ii) if \(aRb\), then \(b'Ra'\), (iii) \((a\wedge b)Ra\), and (iv) not-\(\top R\bot\). For present purposes, the models of interest are those in which \(B\) is a set of `conditions' (essentially predicates) closed under \(\wedge\) and \('\), for which, given a normative system \(\mathcal S\), \(aRb\) represents that \(\mathcal S\) entails that \(a\) implies \(b\). Conditions can be both descriptive and normative. Normative structures often must express `normative correlations' between these types. When the quasi-ordering represents \(\mathcal S\), \(aRb\) describes a normative correlation for \(\mathcal S\) when \(a\) is descriptive and \(b\) is normative. One can also distinguish fragments of a quasi-ordering, and their respective `implication relations', the subrelations of \(R\) on the fragment. Conditions can then be defined for the `joining' of fragments, connections of various forms, especially between the descriptive and the normative parts of a structure, and especially as effected through the use of `intermediary' conditions. These notions are defined precisely in reference to the algebraic concepts introduced, and illustrated through application to simple questions of law.