Conservativeness, Stability, and Abstraction

One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable abstraction principles be strictly logically symmetrically class conservative. The article concludes with an examination of which classes of abstraction principles meet this criteria. 1 Introduction 2 The Technical State of the Art 3 Abstraction and Conservativeness 4 Abstraction Stabilized 5 Characterizing SLSC-Conservativeness 6 Conclusion