Forthcoming Papers Ackermann’s Implication for Typefree Logic

We study the positive fragment of a system of partial logic which has been developed by Ackermann in the course of his typefree foundation of mathematics. One of the prominent features of this system is its implication, which has been supplied with a deductive interpretation according to which A ! B expresses that B is derivable from A. The first section provides Ackermann’s original Hilbert-style formulation of the positive part of his logic and a new formulation using natural deduction. Elaborating Ackermann’s idea concerning the interpretation of!, another new characterization of his positive logic is given in terms of a sequence of consecutive deductive systems. The last section contains a brief account of the algebraic semantics of Ackermann’s system, which has not been considered in the literature up to now.