The Logics of Abstraction

There is a strong interplay between philosophical foundations of mathematics and the logical systems in which they are developed. In the foundational programs in which abstraction principles have been traditionally adopted, i.e. Frege’s logicism and Scottish neo-logicism, such 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; also, abstraction principles can subserve alternative logico-philosophical projects depending on variations of the background logic. In this essay, such an interaction between logic and abstraction is investigated systematically, by showing, on the one hand, how changes in the background logic governing such formal systems can affect the formulation, the mathematical strength and the overall significance of these principles, and, on the other, how different expectations concerning the mathematical or philosophical roles of such principles can suggest changes in the background logic.