Logicism, Neo-Logicism, and the Logics of Abstraction

Abstraction principles play a pivotal role in the foundations of mathematics. In the foundational programs in which they have been traditionally adopted, i.e. Frege’s logicism and Scottish neo-logicism, abstraction principles, conceived as definitions, augment a system of full higher-order logic. Not only, however, does such a logical system raise concerns – both philosophical and technical – for the deployment of abstractive definitions; but, also, abstraction principles can subserve alternative logico-philosophical projects depending on variations of the background logic. We first rehearse the role and significance of definitions by abstraction in logicist and neo-logicist accounts. We then discuss the most compelling technical and conceptual issues related to the logics of abstraction: the debate on the status of higher-order logic; the distinction between predicative and impredicative frameworks, and their relation to paradoxes; the adoption of non-classical logics in contemporary conceptions of abstraction. We conclude by addressing the relations between the logical, the epistemological, and the metaphysical significance of abstraction principles.