An investigation of the propositional calculus used in a particular system of logic

Abstract

This paper contains a summary of some of the results obtained by the author during a study of the propositional part (denoted by A) of a system of free-from-contradiction, type-free logic set up in 1950 by Ackermann (1). It is shown that in its original form the calculus does not possess the desired properties with respect to equivalence of formulae. A calculus A′, which it is shown may be considered as a ‘minimal’ satisfactory extension of A, is constructed. A′ is compared and contrasted with an alternative form A″ of A given by Ackermann in a paper published in 1952 (2). It is proved that A″ is a proper extension of A′. Among the properties of A′ and A″ which are obtained is the resuit that neither calculus possesses a finite complete model. Reference is made to the solution of the decision problem for A, and it is indicated that it is thought that the corresponding problems can probably be solved for A′ and A″. The proofs of many of the results mentioned in the paper are, if given in detail, rather long. In such cases, from space considerations, only outline proofs are given. Complete proofs are contained in (5), which reference also contains several additional properties of the calculi considered.