 Some of the sentences in \(\mathbf{L}\) follow “logically” from others. For example, under the intended interpretation of the truth-functional connectives, \(\alpha\) follows from the sentence \(\alpha\wedge \beta\) and also from the set of sentences \(\{\beta, \beta\rightarrow \alpha \}\). To capture the essentials of deductive consequence, we introduce a  consequence relation, \(\vdash\), which holds between any two sentences \(\alpha \vdash \beta\), whenever \(\beta\) is a deductive consequnce of \(\alpha\) . The consequence operator is assumed to satisfy the following properties, which abstract the characteristic features of deductive logic: