The concepts of lexical compositionality and partial lexical compositionality can now be illustrated in terms of the ILEX template (see Table 2). Immediate Dominance compositionality is represented by the STRUC attribute, and compositional interpretation is indicated by parentheses which represent the application of a semantic or phonetic operator (the first element in the enclosed list) to its operands (the remaining list elements). The notions SEMANTICALLY_LINK and PROSODICALLY_LINK are defined in terms of default unification. The operation of hyphenation is straightforward concatenation of the parts with an intervening hyphen, with the concatenation operation interpreted as a spatial precedence relation. compositionality is defined in general terms for all interpretative features, but each type of interpretation specifies its own operators.
(1) Lexical sign:
(2) Lexical archi-sign:
Table 2: Paradigmatic and syntagmatic inheritance for pussy-willow.
An interesting feature is the operation of lexical insertion, the constraints on which are specified by the `' inheritance type and expressed as attribute paths, i.e. nested AVMs with only one attribute specified per recursion,
In the illustration, the LEMMA pussy-willow paradigmatically inherits properties by default inheritance from the archi-sign compound_noun, and syntagmatically inherits from the head willow `salix' and the modifier pussy `felis'. Those ILEX template properties for pussy-willow which are not specified are completed by unification via inheritance: either percolated up from the head willow or inherited from the archi-sign compound_noun. The LEMMA pussy-willow is seen to be partially rather than fully compositional in that the value for the attribute path INT|MEAN|QUALIA|RELN is specified idiosyncratically. At a higher level in the inheritance path, the value for INT|MEAN|QUALIA|RELN may be specified differently, e.g. as IS_A; the more specific value overrides the more general value.
A lexical sign which inherits all its INT properties from the properties of its PARTS, and its general compositional properties from its CAT attribute (such as function application, concatenation, association), and is not otherwise idiosyncratically specified for INT (i.e. has no default-overrides), is totally compositional.
A lexical sign which inherits none of its INT properties from properties of PARTS, all of these properties being specified idiosyncratically, is totally noncompositional. An extreme example of a sign which is totally non-compositional is a hesitation particle interjection such as `er', i.e. ; however, even this is debatable because the is associated with a flat stylised intonation and together with this intonation has a `phatic' channel-sustaining function.
A lexical sign which inherits some of its INT properties from properties of its PARTS, others being specified idiosyncratically, or which does not inherit compositional properties from the most general subsumer in the inheritance graph, is partially compositional.
The totally compositional and totally non-compositional or idiosyncratic cases are `ideal types' corresponding to absolute or zero adherence to Frege's Principle. Lexical signs, in the general case, exhibit varying degrees of partial compositionality (or, conversely, exceptionality or irregularity), measurable by their depth in the type inheritance hierarchy. The concept of a scale of compositionality applies not just to semantics, but also to surface form.
For example, orthography is partially compositional: in ladies' fingers `okra', the ORTH of the plural fingers is a function of the ORTH of the PARTS finger and s, but the ORTH of the genitive plural ladies' is a more specific function of the PARTS lady and s.
The PHON property is also only partly compositional. The plural appears at first sight to be a general compositional function of the PARTS , namely concatenation (interpreted as temporal immediate precedence: ). However, the compositional function is in fact a more complex morphophonological function which is sensitive to the MANNER and VOICING specifications of the stem-final segment. Morphophonology therefore defines a scale of partial phonetic compositionality.
Perhaps the most interesting cases are the MEAN-SURF parallels in partial compositionality which characterise diachronically lexicalised compounds. For example, the ORTH of dustman is perfectly compositional. The MEAN (in informal terms) is, however, only partially compositional: `municipally employed professional refuse collector', whereby
But the PHON property is also only partially compositional: , and not , i.e. the final consonant of /dst/ is elided and the vowel of /mæn/ is weakened. Partial compositionality of this kind has to be specified idiosyncratically for each lexical item concerned; this is the kind of partial compositionality which, on the diachronic dimension, has led in time to the total non-compositionality of PHON and ORTH with words like woman = (wife,man) or husband = (house,bond).