 We will write \(\Cn(\alpha)\) for the set of all \(\beta\) such that \(\alpha\vdash \beta\). If \(\Delta\) is a finite set of sentences, we write \(\Cn(\Delta)\) for \(\Cn(\wedge_{\alpha\in\Delta}\alpha)\) the set of all deductive consequences of the conjunction of all sentences in \(\Delta\). If \(\Delta\) is an infinite set of sentences, it is a bit more complicated to define \(\Cn(\Delta)\) since the infinite conjunction of all sentences in \(\Delta\) is not a sentence of the formal language. To avoid infinite conjunctions, let \(\alpha_1, \alpha_2, \ldots\) be an enumeration of the sentences in \(\Delta\) and let \(\beta_i = \bigwedge_{j\leq i} \alpha_j\), the conjunction of the first \(i\) sentences in the enumeration. Finally, let \(\Cn(\Delta) = \cup_{i=1}^\infty \Cn(\beta_i)\). It is sometimes convenient to state principles in terms of the consequence operator \(\Cn(\cdot)\). For example, we assume that the deductive consequence relation satisfies the following additional property.