 Propositions enjoy a set-theoretic structure. The relative complement of \(P\), \(\neg P = W\setminus P\), is the set of all worlds in which \(P\) is false. If \(P,Q\) are arbitrary propositions, then their intersection \(P\cap Q\) is the set of all worlds in which \(P\) and \(Q\) are both true. The disjunction \(P\cup Q\) is the set of worlds in which at least one of \(P,Q\) is true. The material conditional \(P\rightarrow Q\) is the set of worlds \(\neg P \cup Q,\) in which either \(P\) is false or \(Q\) is true. If \(P\subseteq Q\) we say that \(P\)  entails \(Q\) and also that \(P\) is  logically stronger than \(Q\). If \(P\subseteq Q\) and \(Q\subseteq P\) we write \(P\equiv Q\) and say that \(P\) and \(Q\) are  logically equivalent. The tautological proposition \(W\) is true in all worlds and the contradictory proposition, the empty set \(\varnothing\), is not true in any world. A set of propositions \(\mathbf{A}\) is  consistent iff there is a world in which all the elements of \(\mathbf{A}\) are true, i.e. if \(\cap \mathbf{A} \neq \varnothing.\) Otherwise, we say that \(\mathbf{A}\) is  inconsistent. A set of propositions \(\mathbf{A}\) is  mutually exclusive iff the truth of any one element implies the falsehood of all other elements. The set of logical consequences of \(\mathbf{A}\), written \(\Cn(\mathbf{A}),\) is the set \( \{ B \subseteq W : \cap \mathbf{A} \text{ entails } B \}\). Note that if \(\mathbf{A}\) is inconsistent, then \(\Cn(\mathbf{A})\) is \(\mathcal{P}(W)\), the set of all propositions over \(W\).