The author wishes to define an implication that should indicate, that \(A \rightarrow B\) if and only if the logical content of \(B\) is part of the content of \(A\). Hence one wants that formula like \(A \rightarrow(\overline A\rightarrow B)\) or \(A\rightarrow (B\rightarrow A)\) should be false in the new calculus. First it is shown that the Hilbert-Bernays calculus of propositions is equivalent to a certain sequential calculus defined by five formulae and seven rules. This sequential calculus is then modified into a calculus with “strong” implication, by adding a formula and severely, weakening the rules. There is a disturbing feature, in so far as the logical rules are no longer universally valid, but different for “distinguished” and “not distinguished” premises. The Hilbert-Bernays system of axioms is then modified accordingly. It is shown that the new system has really the qualities demanded from a strong implication. Finally, one may introduce modalities, by defining as “impossible” those formulae that are disprovable in the new system.