Logemes instead of logics

This paper proposes a phenomenological approach to philosophical logic by reexamining the overlooked yet deeply insightful position of Zaremba in his historical dispute with leading Polish logicians of the early 20th century. Rather than treating this debate as a closed chapter, we argue that Zaremba’s critique remains highly relevant for understanding the limits of formal logic and the need for a more structurally sensitive model of reasoning. His emphasis on the relational and contextual nature of logic challenges the formalist vision of mathematics as a system reducible to axioms. To address this challenge, we introduce a formal mathematical language designed to represent and analyze logemes – the minimal, context-dependent units of logical reasoning. This framework employs simplicial complexes as a topological structure to model how local fragments of reasoning interconnect. Each vertex represents a basic proposition, while higher-dimensional simplices capture the logical relations among them, forming coherent yet flexible substructures. This approach makes it possible to represent interactions among diverse logical systems (such as classical and intuitionistic logic) in a unified, geometrically intuitive manner. By modeling logic not as a unified system but as a dynamic network of interrelated fragments, this work offers a new foundation for philosophical logic – one that is capable of expressing ambiguity, incompleteness, and contextual variation, and that aligns more closely with the actual structure of mathematical and human reasoning. This approach may have a wide and rich range of practical applications in artificial intelligence. It provides a universal representation of databases in the form of logemes, as well as universal models for non-monotonic reasoning. Together, these features make it possible to model key phenomena of consciousness, such as context sensitivity, revision of beliefs, and circularity. By offering a mathematically rigorous framework for non-monotonicity, the approach opens new possibilities for integrating logical, topological, and cognitive structures in AI systems, potentially contributing to more flexible, adaptive, and cognitively plausible models of artificial intelligence.