Our aim is to give a frugal formalization of classical logic based on certain ideas contained in the proof of Hilbert’s Praemissentheorem'', see [1]. An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic. Given two inferential rules, say r and r', we can derive by use of it another rule r(r'). The definition of the operation is due to Pogorzelski [2]. Our attempt is to incorporate the H-operation into the set of usual consequence conditions as given by Tarski. The received proof system might be regarded as a kind of reverse logic’’. The H-operation applies to classical propositional as well as to quantifier logic. But it is not easy to give a natural (that means considered in literature) non-classical system which is closed under the operation.

[1] Hilbert D., Bernays P., Grundlagen der Mathematik, Bd II, Supplement III, Springer Verlag, Berlin 1939.

[2] Pogorzelski W.A. , On Hilbert’s operation on logical rules I, Reports on Mathematical Logic 12(1981), 35-50.