MAKH explains here (1972) why it’s unnecessary to postulate semantic systems that restate grammatical systems...axis makes them already ‘semantic’...The combination of system and structure with rank leads to a fairly abstract grammar (fairly 'deep', in the Chomskyan sense) and enables us to specify fairly accurately in theoretical terms - though not or course in rule-of-thumb terms - just how abstract it is. In principle, a grammatical system is as abstract (is as 'semantic') as possible given only that it can generate integrated structures; that is, that its output can be expressed in terms of functions which can be mapped directly on to other functions, the result being a single structural 'shape' (though one which is of course multiply labelled).
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[1] This is misleading, because it is untrue. On the one hand, this is an extract from Halliday's very early paper which considers arguments for and against the theoretical value of including semantic structure in the model. In the time since then, semantic structure has been part of SFL Theory. Halliday & Matthiessen (1999: 429):
… in our model there are two system-structure cycles, one in the semantics and one in the lexicogrammar. Terms in semantic systems are realised in semantic structures; and semantic systems and structures are in turn realised in lexicogrammatical ones. As we saw in Chapter 6 in particular, 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.
On the other hand, Halliday does not simply postulate 'semantic systems that restate grammatical systems'. Halliday & Matthiessen (1999: 26):
Thus when we move from the lexicogrammar into the semantics, as we are doing here, we are not simply relabelling everything in a new terminological guise. We shall stress the fundamental relationship between (say) clause complex in the grammar and sequence in the semantics, precisely because the two originate as one: a theory of logical relationships between processes. But, as we have shown, what makes such a theory (i.e. an ideation base as the construal of experience) possible is that it is a stratal construction that can also be deconstructed, every such occasion being a gateway to the creation of further meanings which reconstrue in new and divergent ways. Thus a sequence is not 'the same thing as' a clause complex; if it was, language would not be a dynamic open system of the kind that it is.
[2] To be clear, this misunderstands the extract from Halliday (1972). Halliday is here explaining that the system-structure relation (realisation) enables a systemic grammar to be more abstract — where 'more abstract' means 'more semantic' — since system is more abstract than (is realised by) structure.
Importantly, a functional grammar interprets grammatical form (e.g. verbal group) in terms of it function in realising meaning (e.g. Process). Halliday & Matthiessen (2014: 48-9):
We cannot expect to understand the grammar just by looking at it from its own level; we also look into it ‘from above’ and ‘from below’, taking a trinocular perspective. But since the view from these different angles is often conflicting, the description will inevitably be a form of compromise. … Being a ‘functional grammar’ means that priority is given to the view ‘from above’; that is, grammar is seen as a resource for making meaning – it is a semanticky kind of grammar. But the focus of attention is still on the grammar itself. Giving priority to the view ‘from above’ means that the organising principle adopted is that of system: the grammar is seen as a network of interrelated meaningful choices.
