An introduction to mathematical logic covering all the usual topics: compactness and axiomatizability of semantical consequence; Lowenheim-Skolem-Tarski theorems; prenex and other normal forms; and characterizations of elementary classes with help of ultraproducts. Logic is based exclusively on semantics. Truth and satisfiability of formulas in structures are the basic notions, and there is no need to mention logical calculi with axioms and rules (they are the subjects of Volume Two). The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout, and this not just as abbreviations, but in order to gain new insights. These concepts are developed in an introductory chapter which, together with chapters five to nine on equations, can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence as long as Bool