A definition of negation in extended basic logic

E-ISSN： 1943-5886|19|1|29-36

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.19, Iss.1, 1954-03, pp. : 29-36

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

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′.