A diagrammatic proof-theoretic semantics for the Greimas semiotic square

TL;DR

This paper introduces a diagrammatic proof system using spider diagrams to formalise the semiotic square, providing a constructive, rule-based semantics for structural semiotics and non-classical negation.

Contribution

It develops a novel diagrammatic proof-theoretic semantics for the Greimas semiotic square, extending spider diagram calculi to capture semiotic operations and non-Boolean negation.

Findings

Derives the four meta-terms from basic configurations using fixed inference schemas.

Provides a semantic interpretation of the '+' operation as a derivational process.

Shows diagrammatic negation as a zone-determined counter-position, not a Boolean complement.

Abstract

We develop a diagrammatic proof system for a fragment of structural semantics inspired by the Greimas semiotic square, using spider diagrams as the underlying formalism. The basic terms are represented as diagrammatic configurations, and the relations of contradiction and implication are interpreted as transformations governed by a set of inference rules. These transformations are realised as derivations, with proof trees serving as witnesses. Our main result shows that the construction of the four meta-terms can be captured uniformly: each is derivable from a conjunctive pair of basic configurations via a fixed derivation schema composed of contour introduction and habitat transformation rules. This yields a proof-theoretic account of the combinatorial operation underlying meta-term formation, and provides a semantic interpretation of the Greimasian operation `+' as a derivational construction rather than a logical combination. We further show that diagrammatic negation in this setting is not a Boolean complement, but a restricted, zone-determined semantic counter-position, reflecting the relational character of opposition in structural semiotics. The resulting framework provides a compositional, rule-based semantics in which complex configurations are generated constructively from simpler ones. In addition to extending the expressive scope of spider diagram calculi, the system illustrates how diagrammatic reasoning can be used to formalise non-classical semantic operations within a unified inferential setting.