Paraconsistency and Topological Semantics

The well-studied notion of deductive explosion describes the situation where any formula can be deduced from an inconsistent set of formulas. Paraconsistent logic, on the other hand, is the umbrella term for logical systems where the logical consequence relation is not explosive. In this work, we investigate the relationship between some different topological spaces and paraconsistency.

💡 Research Summary

The paper investigates the relationship between paraconsistent logic and topological semantics, offering a systematic construction of two families of modal logics—paraconsistent and paracomplete—based on closed‑set and open‑set topologies respectively. After recalling the classical notion of deductive explosion, the authors define paraconsistency as a non‑explosive consequence relation and motivate the use of topology as a semantic framework. They present two equivalent definitions of a topological space, one emphasizing open sets (σ) and the other closed sets (τ), and recall basic notions such as continuity, openness, and homeomorphism.

In the logical setting, the language is that of unimodal propositional logic with □ and its dual ◇. The □‑operator is interpreted as the interior operator Int, while ◇ is interpreted as the closure operator Clo. To avoid the problem that ordinary negation does not preserve openness or closedness, the authors introduce two new negation symbols: ˙∼, which maps a set to the interior of its complement (open complement), and ∼, which maps a set to the closure of its complement (closed complement). The former yields paracomplete topological models (extensions of propositional variables are open sets), the latter yields paraconsistent topological models (extensions are closed sets). Both logics use the same conjunction and □, but differ in the negation operator.

A key observation is that, because every formula’s extension is forced to be open (or closed), the resulting topologies are necessarily discrete. Proposition 2.3 shows that any two models built on spaces of the same cardinality are homeomorphic, and consequently satisfy exactly the same positive (negation‑free) formulas. The homeomorphism also preserves the special negations, establishing a tight correspondence between the two semantic settings.

The paper then explores how topological properties affect logical behaviour. Connectedness is examined in depth: in a connected space the only subsets with empty boundary are the whole space and the empty set. The authors define “connected formulas” as those that cannot be split into two formulas with disjoint non‑empty extensions. They prove (Proposition 3.2) that every connected formula is satisfiable in some connected classical topological space, and extend this to theories. However, in the paraconsistent setting any connected theory becomes inconsistent because boundary points simultaneously satisfy a formula and its negation (∼). Dually, in the paracomplete setting any connected theory is incomplete because boundary points satisfy neither a formula nor its paracomplete negation (˙∼). These results are formalised in Propositions 3.4–3.7, showing that connected spaces essentially force inconsistency (or incompleteness) for the respective logics.

Continuity and homeomorphisms are further examined. Drawing on earlier work by Kremer and Mints, the authors show that if f : (T, τ) → (T, τ′) is a homeomorphism, then redefining the valuation by V′(p) = f(V(p)) preserves truth of all formulas (Theorem 3.8). This demonstrates that the usual modal‑logic preservation results extend to the paraconsistent/topological framework.

Overall, the paper contributes a clear semantic bridge between paraconsistent (and paracomplete) logics and topology. By introducing specialized negations that respect the open/closed nature of the underlying space, the authors obtain two parallel modal systems. They reveal that the induced topologies are discrete, that connectedness forces inconsistency or incompleteness, and that continuous or homeomorphic maps preserve logical validity. The work opens several avenues for future research, such as extending the analysis to non‑discrete or more exotic topological spaces, investigating algebraic counterparts, and applying the framework to epistemic or temporal paraconsistent logics.