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. Since we allow for a stratification of content systems into semantics and lexicogrammar, we are in a stronger position to construe knowledge in terms of meaning. That is, the semantics can become more powerful and extensive if the lexicogrammar includes systems.It follows then … that for us "knowledge of the universe" is construed as meaning rather than as knowledge. This meaning is in the first instance created in language; but we have noted that meaning is created in other semiotic systems as well, both other social-semiotic systems and other semiotic systems such as perception. Our account gives language more of a central integrative role in the overall system. It is the one semiotic system which is able to construe meanings from semiotic systems in general.Fawcett's model, although in certain ways closer to mainstream cognitive science than ours, is also a systemic-functional model. In other words it is within the same general theoretical framework as that within which our own work is located.
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Fawcett's model is actually a development of Halliday's first theory, Scale and Category Grammar, not Systemic Functional Grammar, as Fawcett (2010) readily admits, and, as a model of syntax, its focus is on structure and form, not system and function. See the close examination of Fawcett's model here.
Fawcett's view of Halliday & Matthiessen (1999) is less than complimentary, and writing on Sysfling on 29 June 2011, he unwittingly disclosed his inability to understand its content:
Indeed, if Michael Halliday and Christian Matthiessen had formed a clear view of the way in which the choices described in their Construing Experience through Meaning determine the choices in the major system networks of the lexicogrammar, they would surely have said so in that book. I have looked hard for a section that makes this connection, but I have yet to find it. This suggests that the model proposed there is simply one possible, half-complete hypothesis that needs to be subject to the normal process in science of development, testing, evaluation, revision (or rejection), retesting, re-evaluation, and so on.