Principia Mathematica description theory: the classical and an alternative notation.
Jordan Howard Sobel · Notre Dame Journal of Formal Logic · 1974
The class of description formulas is not rigorously determined in Principia Mathematical And the syntax of 1-formulas therein countenanced ill-suits their semantics.Correctives for both defects (if the second condition be a defect) are presented below. 1 First, rules for the language of a base theory 9 along with relevant ancillary definitions are set out.Next, these rules are augmented in order to secure the generation of Ί-formulas, and ancillary definitions are amended.Then, a simpler notation is introduced.Next, a theory </? is explained the language of which embraces both notations, and it is shown how they are interchangeable and how the new could indeed be employed as a scaffolding for the introduction of the old.Last, comments are made regarding the semantic misleadingness of the classical notation, and regarding a well-known property of this notation, viz., the term-like behaviour of 'proper' 1-descriptions, that mitigates this misleadingness.These comments are facilitated by the presence in the base theory, and thus in its extension ^?, of 'Fregean' descriptions that are genuine terms.