Stoic and Epicurean Reasoning from the Topological Point of View

This paper proposes a novel approach to Stoic and Aristotelian–Epicurean modal reasoning by replacing traditional Kripkean relational semantics with a topological framework. In the Stoic account, causality and modality are associated with the density of a space, whereas in the Epicurean tradition they are grounded in discreteness and atomic structure. Motivated by this distinction, we argue that causality and modality can be interpreted through the topology of neighborhoods: the Stoic framework corresponds to a density-based notion of logical determinism, while the Aristotelian–Epicurean framework corresponds to a discreteness-based notion of logical contingency. This perspective allows modal operators to be defined via limit-point and interior structures, rather than Kripke's accessibility relations. Also, to capture Stoic logical reasoning, we introduce the notion of logemes, which we model geometrically as simplicial complexes. This representation reflects the flexible, non-linear, and competitive character of ancient inference rules. The proposed topological interpretation not only clarifies the internal structure of Stoic logic but also enables systematic comparison with Aristotelian and Epicurean modal frameworks. In addition, the Stoic interpretation of modality admits concrete applications in data analysis: in particular, it leads to the construction of modal filters that implement a deterministic, topology-based mechanism for noise rejection.