Operadic and Diagrammatic Semantics of the Greimas Semiotic Square

We develop a categorical and operadic semantics for the diagrammatic proof system underlying the Greimas semiotic square. Building on a prior proof-theoretic formulation, we extract a typed signature of diagrammatic inference rules and construct the corresponding free coloured operad OΣ, whose elements correspond to proof trees. This establishes a precise correspondence between diagrammatic derivations and operadic terms, making explicit the compositional structure implicit in the original system. We then interpret these terms as wiring diagrams in a symmetric monoidal setting, yielding a graphical semantics in which intermediate semantic configurations are represented as flows through a network of operations. Within this framework, the construction of the semiotic square is realised as a single composite operation Ω, obtained by operadic substitution of generators corresponding to negation, implication, and meta-term formation. Finally, we consider semantic interpretations of this structure as algebras of OΣ, yielding a category Alg(OΣ), whose morphisms capture structure-preserving translations between interpretations. This provides a formal account of the extensibility of the square across domains such as seme-level analysis, modality, and narratology, and recasts it as a compositional semantic schema rather than a static relational diagram.