Contemporary scientific, technological, and societal challenges increasingly rely on formal methods like logical systems, mathematical structures, and computational reasoning. These formal sciences, however, are themselves undergoing a profound conceptual transformation. Over the last decades, the traditional view that mathematics and logic rest on a single classical foundation has been disrupted by the emergence of rich non-classical alternatives. Intuitionistic, fuzzy, paraconsistent, relevant, non-monotonic, and modal logics each motivate distinctive non-classical set theories. These developments have culminated in two major debates: logical pluralism, which argues that more than one logic is correct, and set-theoretic pluralism, which maintains that there are multiple legitimate universes of sets. Yet, despite their shared motivations and conceptual overlap, the two debates have evolved almost entirely independently.
At the same time, the European research and innovation agenda increasingly emphasizes the need for robust, transparent foundations for disciplines that rely on logic and formal reasoning—from computer science and AI safety to information systems, mathematics education, and the social sciences’ use of formal modelling. The political context—particularly the EU’s strategic interest in trustworthy AI, formal verification, and epistemic resilience—creates an urgent need for better understanding the structures underpinning mathematical and logical reasoning. Foundations matter: they shape what counts as valid inference, acceptable models, or admissible forms of reasoning in scientific and technological domains.
Despite this importance, there is today no unified methodological framework capable of accommodating both classical and non-classical foundations of mathematics, nor is there any systematic account of how logical and set-theoretic pluralism interact. This fragmentation represents a conceptual bottleneck: scientific fields increasingly rely on non-classical logics (e.g. in AI, decision theory, quantum computing), yet the foundational mathematics supporting these logics is either poorly understood or entirely missing. Meanwhile, philosophers lack tools to assess whether arguments in favor of pluralism in logic carry over to pluralism in set theory, or vice versa.
This project addresses this gap.
It proposes the first unified framework capable of generating both classical and non-classical models of set theory from a single algebraic method. This framework will not only demonstrate that major branches of mathematics can be formulated in non-classical settings but will also provide the technical infrastructure needed to compare, evaluate, and understand different foundational paradigms. Building upon this, the project advances a novel integration of logical and set-theoretic pluralism—an unexplored field that promises to reshape parts of philosophical logic and the philosophy of mathematics.
