ABSTRACT
In this chapter, the authors show that Principia Mathematica (PM) is a complete axiomatization of the full Propositional Calculus. 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 well-formed formula (wffs) all the wffs of Propositional Calculus (PC), but unlike PM in having a set of axioms and Rules of Transformation which are not powerful enough to generate every valid wff. 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. A complete axiomatization of the implicational fragment of PC was devised by A. Tarski and P. Bernays and is set out below as the system TB.
