The Dimensionalising Of Semantic Space: Lexicalised Features

Halliday (2008: 180):

More localised features are typically intersected and construed in bundles, usually being assigned some preferred rôle in the configuration of processes: as the process itself, as a participant in the process, as quality of a participant, or as a circumstance (i.e. being assigned to an open word class — in English and Chinese, verb, noun, adjective or adverb). These open word classes construe meanings that we call “lexicalised”; and their “vectors” may be just the presence or absence of some particular component feature.

The Dimensionalising Of Semantic Space: Grammaticalised Features

Halliday (2008: 180):

What we usually call “grammatical” are those sets of features that are construed as single vectors, and which tend to permeate a large number of semantic domains; for example, all potential participants are countable or uncountable, and if countable, are singular or plural.

The Grammatical System Network Theorises The Dimensionalising Of Semantic Space

Halliday (2008: 180):

Thus the lexicogrammar — linguistic form — dimensionalises semantic space; and the grammatical system network theorises this process, just as the phonological system network theorises the dimensionalising of phonetic space.

Polarity & Modality As Pre-Metafunctional

Halliday (2008: 180n):

polarity is as it were pre-metafunctional: it develops (and probably evolved) prior to the distinction between ideational and interpersonal meanings. (Rather in the same way that, in the physical world, positively and negatively charged particles seem to be prior to the distinction between matter and information!) It may be that the “energy levels” of  modality (high / median / low) are also of this kind: both modality and polarity are regularly grammaticalised in combination with other features.

Protolanguage, Appraisal & Grammaticalisation

Halliday (2008: 179-80):

In terms of children’s early language development, the interpersonal metafunction provides the prototype of how meanings come to be grammaticalised. The two systems that were first grammaticalised by one small child (Nigel, aged 0;10) were:

polarity: positive / negative 
modality: value: low / high

followed shortly by the two forms of “appreciation” in conjunction with the feature “positive”:

appreciation (positive): impact (“that’s interesting”) / quality (“that tastes nice”)

Note that these were not yet mother tongue; they were protolanguage, realised by sounds and gestures. But they were systemic, or at least proto-systemic; and they provided the model for the linguistic systems of appraisal, where each lexical item realises the intersection of an appraisal feature with polarity and/or modality.

Appraisal: Attitude

Halliday (2008: 179):

This is a grammatical system that is realised by a selection of lexical items. Each such item is uniquely identified as a set of intersecting grammatical features; eg complicated is

appraisal: attitude: appreciation: composition: ( complexity : complex /  polarity: negative …)

as well as other general grammatical features (e.g. as distinct from confusing, which is “effective”, as in it confuses me, complicated is “descriptive”).  Note that I am interpreting the feature "undesirable" (the "snarl" member of the "purr/snarl" opposition) as negative in the environment of the interpersonal.