 Another system worthy of consideration is the trivalent logic of “conditional objects” investigated by Dubois and Prade (1994). This system is itself closely related to a three-valued treatment of conditionals presented in Adams (1986). The system rests on de Finetti’s original table, but only formulae from the flat fragment \(L_1\) are admitted. Validity in Dubois and Prade’s approach is defined as follows. Assume \(\Gamma\) is a set of flat conditionals of the form \(A_i&gt;B_i\). Then define \(\Gamma \models A&gt;B\) iff either \(A\) classically entails \(B\) or for every trivalent valuation the value of the quasi-conjunction of some subset of conditionals in \(\Gamma\) is less than or equal to the value of \(A&gt;B\) (for the usual ordering of truth-values: \(0&lt;½&lt;1\)). Quasi-conjunction—a connective proposed by Cooper (1968: 305) and in a different guise by Adams (1966: 306), also proposed by Belnap (1973:60), and earlier by Sobociński 1952 in relation to a different trivalent conditional —differs from min conjunction in that the conjunction of classical values is classical, and the conjunction of ½ with a classical value is that classical value. Quite remarkably, this relation of logical consequence can be axiomatized by the system P of preferential entailment of Kraus, Lehmann and Magidor, which is discussed in detail below. Thus, the associated notion of consequence does not support Monotonicity, Transitivity, and Contraposition of the conditional (see  section 3.2).