A definition of negation in extended basic logic

In a previous paper it was shown that the system K of basic logic could be formulated in a simpler way owing to the fact that the proper ancestral could be defined in terms of the other concepts of that system. In the present paper analogous but more far-reaching results will be obtained for the system K′ of extended basic logic. In particular we will show that negation and the dual of the proper ancestral, as well as the proper ancestral itself, are definable in terms of the other concepts of K′. Hence, in order to define K′, we need to add only a single non-finitary rule to the rules used to define K. This rule was already among the rules originally used to define K′. It asserts that ‘Aa’ is in K′ if (and only if) every ‘b’ is such that ‘ab’ is in K′.We will also show that a large class of non-finitary classes and relations are represented in K′, among which is K′ itself, just as all finitary syntactical classes and all finitary two- and three-place syntactical relations are represented in K, one of which is the finitary syntactical class K itself. The point is that K′ is adequate to handle the sort of transfinite induction that is essential in formulating K′, just as K is adequate to handle the ordinary finitary mathematical induction required in defining K′.