The three dimensions of a formal modeling of the language: syntagmatic, paradigmatic and semiotic

We show that every set of productions of every natural language can be described as a three-dimensional space: the syntagmatic axis is the axis of combinations of signs within the text, the paradigmatic axis is the axis of commutations in each position in the text, and the semiotic axis is the axis internal to the signs, that links the signified to the signifier, the meaning to the text. Every formal modeling of a natural language has to cover this 3-dimensional space. Moreover, in order to capture the regularities, a model seeks after covering the largest possible parts of this space with each rule, which is difficult due to the mutual dependency of the three dimensions. In order to better understand the issues, we will compare two strategies illustrated by a well-known model, LTAG, and another similar one, MTUG, which is inspired by the former but favors a cutting along the syntagmatic and semiotic axes.