Summary: Logicians treat assertions as true, believed or merely hypothesized sentences. The reasoner who uses them, however, is the sole referee who can validate their truth, their aptness to describe an actual situation, their strength (as beliefs) or the relevance of their use in the current logical context. Moreover, the reasoner actively counts on these factors, as part of the reasoning process itself, and should normally be capable, when asked to do so, to assign consistently relative strengths to the assertions used. The paper assumes, first, that assertions have -- each -- an associated, measurable strength, and that, second, this strength has significant -- and measurable -- effects on the truth of the sentences, the validity of the conclusion and the soundness of the reasoning. The concepts and formulas required for this are explored, and a semantics and proof theory for a sentential calculus of assertions are proposed as a natural extension of ordinary two-valued reasoning. The resulting theory, though reminiscent of probability, is autonomous, self-contained and of a purely logical nature.