The Logic of Abstraction

Abstractionist programs in philosophy of mathematics are foundational projects based on theories including logic and one or more abstraction principles. Most of these programs are inspired by Freges Logicism and share the theses of this seminal project, namely the aprioricity of arithmetic and the Platonistic nature of its realm. In this project, I will focus on the formal contribution and philosophical import of the logical background to these programs, in order to explore non-classical revisions. This investigation aims to analyze the critical problems regarding the standard (namely, classical logic-based and philosophically Fregean) abstractionist programs. Classical logic plays a crucial even if little-known role both in the derivation of every abstractionist paradox and in the failure of current criteria of logicality by abstract entities. The aim of the preliminary stage is to investigate the traditional choice of classical logic, in order to show that it is deeply related to the philosophical goal of Fregean projects. On the one hand, classical logic offers a solid background for the full referentiality of the language, which, in turn, is a necessary condition to support the aprioricity of the abstracts. On the other hand, classical logic semantics includes a standard denotation function, which, in turn, is required to uphold a Platonist metaphysics. Combining all these suggestions will highlight that while there are no strong formal reasons, there are strong philosophical motivations to prefer classical logic, given a Fregean and Platonist vocation. Such a condition suggests focusing on the formal results and, accordingly, exploring a less-known philosophical path: on the one hand, challenging the adoption of classical logic allows us to deal with paradoxes and logicality; on the other hand, renouncing classical logic implicitly requires a neutral philosophical view on the abstraction, that is opened to epistemic and metaphysical alternatives to Fregean Platonism. Therefore, the overarching aim of this project consists in questioning the implicit assumption that classical logic is the best logical framework for abstractionist programs. The research will be driven by the following research questions. The first question asks what are the necessary conditions that the abstraction principles and the logical axioms must fulfill in order to achieve mathematical and philosophical results. The second question investigates whether and how the adoption of a non-classical logic could provide a partial equivalence relation on the right-hand side of the abstraction principles that is able to avoid their paradoxical effects and improve their mathematical consequences. The third question asks whether and how the adoption of a non-classical logic could be compatible with a less demanding interpretation of the abstraction principles and could have an impact on their semantical properties. Finally, the abstractionist project arising from the adoption of a non-classical logic background will be analyzed through the lens of Deflationism. Such a novel approach liberates abstractionism from the philosophical commitment to Platonism and allows a more genuine comparison between Logicist and Structuralist abstraction.