 If you think of \(X\) as a set of claims, then the first condition tells you that the consequences of a set of claims includes the claims themselves. The second condition demands that the consequences of \(X\) just are the consequences of the consequences of \(X\). Both of these conditions can be motivated from reflection on the model-theoretic and proof-theoretic approaches; and there are other such conditions too. (For a general discussion, see the entry on  algebraic propositional logic.)  But as with many foundation issues (e.g., ‘what are the essential features of consequence relations in general?’), even such minimal conditions are contentious in philosophical logic and the philosophy of logic. For example, some might take condition (2) to be objectionable on the grounds that, for reasons of vagueness (or more), important consequence relations over natural languages (however formalized) are not generally transitive in ways reflected in (2). (See Tennant 1994, Cobreros et al 2012, and Ripley 2013, for philosophical motivations against transitive consequence.) But we leave these issues for more advanced discussion.