 Given a logical deduction system \(H\), the relation \(\vdash\) defined by \(X \vdash a\) if \(a\) is deducible from \(X\) in \(H\) is (according to all we have already seen) a finitary consequence relation. Nonetheless, we are used not only to syntactic definitions of consequence relations but also to semantic definitions. For example, we define classical propositional consequence using truth valuations, first-order consequence relation using structures, intuitionistic consequence relation using Kripke models, etc. Sometimes these model-theoretic definitions of consequence relations define non-finitary consequence relations, for example, the consequence relations for infinitary formal languages and the consequence relation of second-order logic with the so-called standard semantics.