… the development of the Cardiff Model of language and its use (although our main focus, like Halliday's, has remained on language — and in our case on the cognitive-interactive modelling of language and its use).
… Halliday's first tentative explorations of his theoretical shift (in Halliday 1966), from treating system networks as choices at the level of form to treating them as choices at the level of meaning.
… a series of Halliday's descriptions of areas of English grammar, ranging from his early system networks (from Halliday 1964), which were of course conceived of as being at the level of form, to later descriptions, some of which illustrate the concept that they can be interpreted as being choices between semantic features. …
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[1] This is misleading, since it implies that Halliday's theory is not concerned with a "cognitive-interactive modelling of language and its use". The difference lies in how these dimensions are understood. Halliday understands the interactive dimension of language as the interpersonal metafunction, understands use in terms of variation along the cline of instantiation, and understands cognition in terms of meaning. In their work subtitled A Language-based Approach to Cognition, Halliday & Matthiessen (1999: ix-x) write:
It seems to us that our dialogue is relevant to current debates in cognitive science. In one sense, we are offering it as an alternative to mainstream currents in this area, since we are saying that cognition "is" (that is, can most profitably be modelled as) not thinking but meaning: the "mental" map is in fact a semiotic map, and "cognition" is just a way of talking about language. In modelling knowledge as meaning, we are treating it as a linguistic construct: hence, as something that is construed in the lexicogrammar. Instead of explaining language by reference to cognitive processes, we explain cognition by reference to linguistic processes. But at the same time this is an "alternative" only if it is assumed that the "cognitive" approach is in some sense natural, or unmarked.
[2] This is misleading. The stratal distinction of form vs meaning is the distinction in Fawcett's model, but never in Halliday's. Even when Halliday did propose a level of form (Halliday 1961), the distinction was substance vs form vs situation. In this early model, the analogue of "meaning" was termed 'context' and construed as an interface between form and situation:
However, by 1976, if not before, Halliday had stratified language in its current formulation. Halliday & Hasan (1976: 5):
[3] To be clear, in Halliday (1978), the systems of transitivity, mood and theme were construed as semantic systems that specified different metafunctional structures that were mapped onto the clause. However, this model was soon reconstrued, largely due to the need to systematically account for grammatical metaphor as an incongruence between semantic selections and lexicogrammatical selections. As Halliday & Matthiessen (1999: 429) later explained in their description of semantic systems:
… grammatical metaphor is a central reason in our account for treating axis and stratification as independent dimensions, so that we have both semantic systems and structures and lexicogrammatical systems and structures.
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For more of Fawcett's misrepresentations of Halliday's theorising, see the clarifying critiques here.