 Section 1 introduces the idea of formal sentential languages as (absolutely free) algebras, their primitive connectives having the status of the fundamental operations of these algebras, and explains the notion of a consequence relation on such a language. At a more general level, Galois connections are also introduced, and a simple observation made concerning what we call binary relational connections, to be applied in later sections. Section 2 introduces the fundamental idea of a valuation for a language (a bivalent truth-value assignment to its formulas). Truth-functionality and a generalization thereof called pseudo-truth-functionality (of a connective with respect to a class of valuations) are explained, as well as the idea of sequents and sequent-to-sequent rules. We then move to a more general conception of sequents as originally conceived, allowing a set of formulas to play the conclusion role rather than just a single formula, as well as to the associated idea of generalized consequence relations, emphasizing Carnap’s motivation for making this generalization. That motivation can be summarised with reference to one connective. We take negation as our example. Consider the consequence relation determined by the class of all valuations on which the usual negation truth-function connective is associated with the negation connective. This can be regarded as the classical logic of negation, at least if logics are identified with consequence relations. Now, there are additional valuations consistent with this consequence relation on which the negation truth-function is not associated with the negation connective. (The terminology of a valuation’s being consistent with a consequence relation is adapted from Dana Scott. A precise definition appears in Section 2.) But when we consider instead generalized consequence relations, the analogous discrepancy does not arise. Section 2 includes a statement of a result of D. Gabbay which isolates precisely the truth-functions for which such discrepancies do not arise (roughly: generalizations of the conjunction truth-function and various projection functions), as well as of a result of W. Rautenberg showing that despite this expressive weakness in the apparatus of consequence relations, they are sufficiently discriminating—without the need to pass to generalized consequence relations—to distinguish the logical properties of any two truth-functions (of the same arity).