Syntactic Concepts. We work with an arbitrary but fixed sentential language \(L\), for which sequents are pairs \(\langle\Gamma,A \rangle\) with \(\Gamma \subseteq L\), \(A\in L\). An \(n\)-premiss rule \(\rho\) is an \((n+1)\)-ary relation on the set of sequents. If \(\langle \sigma_0, \dots, \sigma_n \rangle\in\rho\) then \(\sigma_n\) is the conclusion-sequent and \(\sigma_0, \dots, \sigma_{n-1}\) the premiss-sequents for the application \(\langle \sigma_0, \dots, \sigma_n \rangle\) of \(\rho\). A rule \(\rho\) is derivable from a set of rules \(R\) if every set of sequents closed under each rule in \(R\) is closed under \(\rho\). Structural rules are understood in the sense of Gentzen: as rules whose applications consist of all instantiations of a sequent-to-sequent schema in which no connective of \(L\) appears (by contrast with operational rules). Closure under certain structural rules is necessary and sufficient for a set of sequents to constitute a consequence relation; these are the rules \((\mathbb{R})=\) the zero-premiss rule \(\{ \langle \{A\}, A\rangle \mid A\in L\}\), \((\mathbb{M}) =\) the one-premiss rule \(\{\langle \langle \Gamma, A\rangle, \langle \Delta,A \rangle\rangle \mid \Gamma \subseteq \Delta \subseteq L\}\), and the cut or transitivity rule \((\mathbb{T})\) which we will further specify here. Let \(S= \{(\mathbb{R}), (\mathbb{M}), (\mathbb{T})\}\). Semantic Concepts. A valuation (for \(L)\) is a mapping from the set of formulas (of \(L)\) to the two element set \(\{T,F\}\) of truth-values. A sequent \(\sigma= \langle \Gamma,A \rangle\) holds on a valuation \(v\) if \(v(C)=T\) for all \(C\in\Gamma\) implies \(v(A)= T\). For a class \(V\) of valuations, we say sequent \(\sigma\) is \(V\)-valid if \(\sigma\) holds on each \(v\in V\). The local range, \(\text{Loc} (R)\), of a collection of rules \(R\) is the set of all valuations \(v\) such that for any \(\langle \sigma_0, \dots, \sigma_n \rangle\in\rho \in R\) if each of \(\sigma_0, \dots, \sigma_{n-1}\) holds on \(v\), then \(\sigma_n\) holds on \(v\). The global range, \(\text{Glo} (R)\), of a collection \(R\) of rules is the class of sets \(V\) of valuations such that for any \(\langle \sigma_0, \dots, \sigma_n \rangle \in\rho \in R\) if each of \(\sigma_0, \dots, \sigma_{n-1}\) is \(V\)-valid then \(\sigma_n\) is \(V\)-valid. The local and global consequence operations \(LCn\) and \(GCn\) are closure operations defined on the set of rules by: \[ LCn (R) = \biggl\{\rho \mid \text{Loc} (R) \subseteq \text{Loc} \bigl(\{\rho\} \bigr) \biggr\}, \quad GCn (R) = \biggl\{\rho\mid \text{Glo} (R) \subseteq \text{Glo} \bigl(\{\rho\} \bigr) \biggr\}. \] Connections. The main connection between the above syntactic and semantic concepts is given by: \(\rho\in GCn(R)\) if and only if \(\rho\) is derivable from \(R \cup S\). By contrast, the local consequences of a set of rules \(R\) are not guaranteed to be derivable from \(R\) with assistance of the structural rules \(S\). (Examples are given.) Reference. The role of the `global' concepts here is a vindication of the emphasis placed on them in a related context by \textit{J. W. Garson} [``Categorical semantics'', in: J. M. Dunn and A. Gupta (eds.), Truth or consequences: Essays in honor of Nuel Belnap, 155-175, Kluwer, Dordrecht (1990)].