there is also Teun van Dijk's ongoing and in many respects well motivated critique of the SFL notion of context, which reflects a lot of frustration with the SFL approach and its lack of engagement with certain issues, such as cognition:
Discourse and Context A Sociocognitive Approach 2009.
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See the 600+ page magnum opus:
Halliday MAK and Matthiessen CMIM 1999 Construing Experience Through Meaning: A Language-Based Approach To Cognition
From the Preface (ix-x):
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. It seems to us that current approaches to neural networks, "connectionist" models and the like, are in fact more compatible with a semantic approach, where "understanding" something is transforming it into meaning, and to "know" is to have performed that transformation. There is a significant strand in the study of language — not only in systemic functional theory but also for example in Lamb's relational networks — whereby "knowledge" is modelled semiotically: that is, as system-&-process of meaning, in abstract terms which derive from the modelling of grammar.
Moreover, contrast the above with the theoretical perspective of Van Dijk in Discourse and Context A Sociocognitive Approach (2014: 281):
Note, though, that since the empiricist Functional Systemic framework explicitly does not have a cognitive dimension (Van Dijk, 2008a), the analysis is presented in terms of meaning ‘shortcuts’ as allowed by the grammar of English, not really in terms of tacit or implicit knowledge as it is usually dealt with in philosophy or psychology.
Given that Van Dijk is theorising in a different tradition with different assumptions from SFL — most significantly: the separation of meaning and knowledge — it is not surprising that he would expect 'context' to encompass different phenomena, but it is poor (and misleading) scholarship not to recognise that fact in any critique. Accordingly, Bateman's unsupported claim that any such critiques are 'in many respects well motivated' is not itself "well motivated".
Postscript
Bateman's claim that van Dijk's (2009) criticism of the SFL notion of context is "well motivated in many respects" does not withstand close scrutiny. Van Dijk's argument is basically that the SFL model needs reworking because he does not understand the metafunctional dimensions on which it is organised. In terms of scholarship, Van Dijk did not bother to consult any edition of IFG, or even Halliday & Matthiessen's Construing Experience Through Meaning: A Language-Based Approach To Cognition.
In truth, Van Dijk's motivation is merely ideological. He misconstrues SFL Theory as anti-cognitive, and this he conflates with behaviourism.