Periodic Reporting for period 1 - UCAN (Unifying classicality and non-classicality)
Berichtszeitraum: 2023-06-01 bis 2025-12-31
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.
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.
Activities Performed and Main Scientific Achievements:
During the reporting period, the UCAN project has focused on the development of a unified framework for the study of classical and non-classical logics, based on algebra-valued models. The main activities carried out include:
(1) Systematic development of algebra-valued models for non-classical logics.
(2) Formal investigation of logical pluralism by means of algebra-valued models.
Publications and on-going work:
Articles:
Jockwich S., 202Xa, Maximizing the class of non-classical models of ZFC, manuscript.
Jockwich S., 202Xb, On Logical and Set-theoretic Pluralism, manuscript.
Jockwich, Venturi and Tarafder, The internal modal logic of forcing, manuscript.
Jockwich S., et. al. (Venturi and Tarafder), 2024, ZF and its interpretations, Annals of Pure and
Applied Logic.
Jockwich S., 2025a, A model of connexive set theory, Studia Logica.
Jockwich S., 2025b, Logical Pluralism via Mathematical Convergence, Erkenntniss.
Jockwich, Venturi and Tarafder, (Forthcoming) On the Axiom of Choice in a class of
paraconsistent non-classical models of set theory, Logic Journal of the IGPL.
Book chapter:
Jockwich S., 2026 (Forthcoming), Mereological Forcing, Logic trends in Asia (series)
Book:
Jockwich et al. (Venturi and Tarafder), 202X, Non-classical model of set theory and Independence
proofs. Submitted.
During the reporting period, the UCAN project has focused on the development of a unified framework for the study of classical and non-classical logics, based on algebra-valued models. The main activities carried out include:
(1) Systematic development of algebra-valued models for non-classical logics.
(2) Formal investigation of logical pluralism by means of algebra-valued models.
Publications and on-going work:
Articles:
Jockwich S., 202Xa, Maximizing the class of non-classical models of ZFC, manuscript.
Jockwich S., 202Xb, On Logical and Set-theoretic Pluralism, manuscript.
Jockwich, Venturi and Tarafder, The internal modal logic of forcing, manuscript.
Jockwich S., et. al. (Venturi and Tarafder), 2024, ZF and its interpretations, Annals of Pure and
Applied Logic.
Jockwich S., 2025a, A model of connexive set theory, Studia Logica.
Jockwich S., 2025b, Logical Pluralism via Mathematical Convergence, Erkenntniss.
Jockwich, Venturi and Tarafder, (Forthcoming) On the Axiom of Choice in a class of
paraconsistent non-classical models of set theory, Logic Journal of the IGPL.
Book chapter:
Jockwich S., 2026 (Forthcoming), Mereological Forcing, Logic trends in Asia (series)
Book:
Jockwich et al. (Venturi and Tarafder), 202X, Non-classical model of set theory and Independence
proofs. Submitted.
The essence of UCAN was to develop a genuinely new common framework in which both classical and non-classical foundations of mathematics can be studied side by side. The project was designed to make precise — in a mathematically rigorous way — how different logics and different conceptions of sets can coexist, interact, and illuminate one another. In this sense, UCAN connected two major philosophical debates: logical pluralism and set-theoretic pluralism.
Result 1: A common arena for classical and non-classical foundational theories
UCAN achieved this result by developing a powerful generalisation of algebra-valued models based on alternative ways of interpreting the fundamental notions of membership (“being an element of”) and identity (“being the same object”). This made it possible, for the first time, to construct quotient models of full Zermelo–Fraenkel set theory (ZFC) that are paraconsistent—that is, they can accommodate controlled inconsistencies—while still closely resembling the familiar classical universe of sets. This is a significant and novel contribution both to paraconsistent set theory and to set-theoretic foundations more broadly.
We then extended this methodology to a wide class of algebraic structures, including those that correspond to connexive logics. Although connexive logics depart sharply from classical logic—especially in the strong principles they impose on implication and negation—they nevertheless give rise, within our framework, to set-theoretic universes that are strikingly similar to the classical one. This led to the first connexive model of ZFC, filling a major gap in the foundations of connexive logic, where no comparably expressive mathematical universe had previously been available.
Finally, the same framework was successfully extended beyond set theory to mereology, the theory of parts and wholes. This resulted in the development of mereological forcing, a new analogue of set-theoretic forcing that provides a systematic method for studying independence and compatibility phenomena in mereological theories. In this way, UCAN created a unified semantic arena in which both classical and non-classical foundational theories of sets and parts can be rigorously explored and compared.
Result 2: A robust defense of pluralism in the foundations of mathematics.
Beyond its technical achievements, UCAN delivered a substantial conceptual and philosophical contribution. By demonstrating how many different logics and many different set-theoretic universes can be realized within a single, unified mathematical framework, the project provides a concrete and principled form of pluralism in the foundations of mathematics.
Within the algebra-valued setting developed in UCAN, different choices of underlying algebra generate different logics and different mathematical universes, while preserving a common structural core. This makes it possible to compare classical, paraconsistent, intuitionistic, fuzzy, quantum, and connexive foundations in a precise and controlled way, rather than treating them as disconnected or competing alternatives. As a result, UCAN shows that foundational diversity is not a sign of inconsistency or weakness, but a systematic and intelligible feature of mathematics itself.
In this way, UCAN offers a rigorous framework for understanding how multiple logics and multiple foundational theories can coexist within a single overarching architecture — not as rivals to be eliminated, but as complementary and mutually illuminating perspectives on mathematical reality.
Result 1: A common arena for classical and non-classical foundational theories
UCAN achieved this result by developing a powerful generalisation of algebra-valued models based on alternative ways of interpreting the fundamental notions of membership (“being an element of”) and identity (“being the same object”). This made it possible, for the first time, to construct quotient models of full Zermelo–Fraenkel set theory (ZFC) that are paraconsistent—that is, they can accommodate controlled inconsistencies—while still closely resembling the familiar classical universe of sets. This is a significant and novel contribution both to paraconsistent set theory and to set-theoretic foundations more broadly.
We then extended this methodology to a wide class of algebraic structures, including those that correspond to connexive logics. Although connexive logics depart sharply from classical logic—especially in the strong principles they impose on implication and negation—they nevertheless give rise, within our framework, to set-theoretic universes that are strikingly similar to the classical one. This led to the first connexive model of ZFC, filling a major gap in the foundations of connexive logic, where no comparably expressive mathematical universe had previously been available.
Finally, the same framework was successfully extended beyond set theory to mereology, the theory of parts and wholes. This resulted in the development of mereological forcing, a new analogue of set-theoretic forcing that provides a systematic method for studying independence and compatibility phenomena in mereological theories. In this way, UCAN created a unified semantic arena in which both classical and non-classical foundational theories of sets and parts can be rigorously explored and compared.
Result 2: A robust defense of pluralism in the foundations of mathematics.
Beyond its technical achievements, UCAN delivered a substantial conceptual and philosophical contribution. By demonstrating how many different logics and many different set-theoretic universes can be realized within a single, unified mathematical framework, the project provides a concrete and principled form of pluralism in the foundations of mathematics.
Within the algebra-valued setting developed in UCAN, different choices of underlying algebra generate different logics and different mathematical universes, while preserving a common structural core. This makes it possible to compare classical, paraconsistent, intuitionistic, fuzzy, quantum, and connexive foundations in a precise and controlled way, rather than treating them as disconnected or competing alternatives. As a result, UCAN shows that foundational diversity is not a sign of inconsistency or weakness, but a systematic and intelligible feature of mathematics itself.
In this way, UCAN offers a rigorous framework for understanding how multiple logics and multiple foundational theories can coexist within a single overarching architecture — not as rivals to be eliminated, but as complementary and mutually illuminating perspectives on mathematical reality.