chapter  20
6 Pages

The System TB

WithG. E. Hughes, D. G. Londey

We have seen that PM is a complete axiomatization of the full Propositional Calculus. Now there are two ways in which a system could fail to be complete in this way. On the one hand, the system could be like PM in having as its set of wffs all the wffs of PC, but unlike PM in having a set of axioms and Rules of Transformation which are not powerful enough to generate every valid wff. (An example of such a system would be PM with, say, A2 deleted.) On the other hand, the system could be unlike PM in having as its wffs only some selection from the wffs of PC, but like PM in having a set of axioms and Rules of Transformation which are powerful enough to generate every one of its wffs which is valid. Although such a system would not be a complete axiomatization of the whole of PC, it would be a complete axiomatization of a certain part of PC. We shall call such a part of PC a fragment of PC, and an axiomatization of a fragment might be called a fragmentary axiomatization of PC.