The Semiotic Square

Greimas's semiotic square is a tool for mapping the logical structure of semantic oppositions. Given two contrary terms $S_1$ and $S_2$, the square generates their contradictories $\neg S_1$ and $\neg S_2$, producing a four-term structure with six typed relations: contradiction, contrariety, sub-contrariety, and implication (deixis).

Relations

The four relation types capture distinct logical constraints:

• Contradiction: mutually exclusive truth values. $S_1 \leftrightarrow \neg S_1$, where one must hold and the other must not.
• Contrariety: cannot both be true. $S_1$ and $S_2$ may both fail, but cannot both hold.
• Sub-contrariety: cannot both be false. $\neg S_1$ and $\neg S_2$ may co-exist, but at least one must hold.
• Implication: directed deixis. $S_1 \to \neg S_2$ and $S_2 \to \neg S_1$ define the diagonal channels.

Klein Four-Group

The group view encodes the four corners as two bits and treats moves on the square as composable bit-flips:

Generator $a$ flips the first bit (contradiction axis), $b$ flips the second (contrariety axis), and $ab$ flips both (diagonal). Each generator is its own inverse, so applying it twice returns to the origin, making the group an involution lattice.

Flows as Reachability

The animated particles trace reachability through the enabled structure. In “from selected node” mode, particles emit from the chosen corner along directed and optionally undirected edges, visualizing which positions are accessible from a given semantic commitment. The random walk mode reveals which corners become steady-state attractors under the current edge configuration.

Notes

• The formal structure (edges, types) is intentionally separated from interpretation (what $S_1$, $S_2$ mean).
• Try concrete labels (e.g., Legal/Illegal) and compare presets.
• In group view, repeatedly apply $a$, $b$, $ab$ to verify involution and commutation.
• The adjacency matrix in settings shows the enabled structure as a typed graph.