KRYSIA BRODA MARCELLO D’AGOSTINO MARCO MONDADORI ∗ A SOLUTION TO A PROBLEM OF POPPER ABSTRACT: Popper’s logical systems were an attempt to “reform the teaching of elementary logic”, but were regarded by their own author as “bad and ill-fated”. However, the problem raised by Popper was, and still is, a real one. In fact we show that the popular approaches in terms of natural deduction and analytic tableaux are unsatisfactory. Then we present a system of classical logic which combines the virtues of both and can be generalized to a wide range of non-classical logics. 1. INTRODUCTION Between 1947 and 1948 Karl Popper published a series of papers 1 on mathematical logic whose main intention was “to simplify logic by de- veloping what has been called by others ’natural deduction”’ 2 in an at- tempt “to reform the teaching of elementary logic”. 3 These works were also motivated by a second problem, namely the problem of finding a general definition of the notions of logical word and logical validity. We shall restrict ourselves to the first problem, the simplification of elemen- tary logic and the reform of its teaching. Popper called his own papers “bad and ill-fated”. 4 However, the problem of “simplifying” logic was a real one. Our claim is that, in spite of many advances, the question is not quite fully settled. We do not endorse the received view that natural deduction systems or the method of analytic tableaux 5 provide a satisfactory solution. We shall criticize both types of systems on the assumption that classical logic is the best starting point as far as the teaching of elementary logic is concerned. (We shall not defend this assumption here, since it seems to be shared by all introductory textbooks currently on the market.) We shall then present a system of classical logic which, in our view, supersedes both natural deduction and the tableau method, providing a better solution to Popper’s problem. As we shall argue, the main drawback of natural deduction is that its rules do not capture the classical meaning of the logical operators. 147