A logic is a class of statements that all follow from a given set of statements. (The latter are known as ‘premises’ or ‘axioms.’) And logic is the study of such statement-sets. A logic, in the mathematical or formal sense, is a class of statements that follows, in accordance with purely syntactically (formally) defined rules of inference, from a given set of statements. And mathematical logic is the study of such statement-sets. Equivalently, mathematical (or ‘formal’ or ‘symbolic’) logic is the study of recursively defined statement-sets, a ‘recursion’ being a mathematical function that assigns statements to statements. With regard to your question, when logic is done in a purely intuitive, ad hoc way, it is not a science: it is more a branch of philosophy that is proto-scientific, not genuinely scientific, in nature. When logic is done mathematically—-when the rules of inference are defined formally, and therefore clearly, and when, consequently, judgments are made on mathematical, as opposed to purely intuitive grounds—then logic is a science, but with the heavy qualification that it is not an empirical science. In other words, when logic is a science, it is a science in the sense in which pure mathematics is a science (indeed, it is itself a branch of pure mathematics), meaning that it is based on purely conceptual inter-relations among statements and, unlike biology or psychology or physics, is to no degree dependent for the validity of its pronouncements on considerations of experience or observation.