Traditionally, the linguistic structure of signs is characterised in terms
of three basic notions:
level of representation (abstraction, description);
syntagmatic relation; paradigmatic relation.
The level of representation (abstraction, description etc.), includes
compositional levels of morphology, syntax, text, and interpretative levels
of semantics, phonetics. At each level, structure is further defined by
syntagmatic relations, including concepts of dependency,
valency and headedness, and by
paradigmatic relations, including concepts of markedness.
Syntagmatic relations are part-whole and part-part
relations and paradigmatic relations are
similarity relations which define classes of linguistic units and
oppositions between sub-classes. These notions will be characterised in
more detail below.
Level of representation (abstraction, description etc.): A coherent set of descriptive categories together with methodological criteria and formal representation devices for these categories. Levels are assigned to a scale of well-defined ranks corresponding to linguistic levels of description from phoneme-like units through morpheme-like units, simple, derived and compound words, phrases, sentences (including idioms and proverbs) to ritualised exchanges. At each rank a distinction between lexical and nonce (ad hoc) items is defined, and the rank scale of lexical items constitutes an idiomaticity hierarchy. The word is a basic rank in the sense of Rosch's notion of basic category [Rosch 1978].
A distinction is made at each rank between signs and their co-interpretation in terms of phonetic and orthographic surface form and meaning. The duality of co-interpretation, shared by many linguistic theories, explicates the traditional semiotic triangle in terms of a sign for which there exists on the one hand a model of surface form (sound or writing, gesture, scent etc.), and on the other hand a model of situational meaning. Whether the sign and its two types of interpretation are assigned cognitive (conceptual, mentalistic) interpretations in addition to the behavioural and observational criteria for surface (and, in part, semantic) interpretations is more a question of a linguist's epistemological stance than of direct empirical consequence.
The pair of interpretation functions co-interprets items at different ranks
such as the phoneme, the morpheme,
the word, the sentence, the turn or
dialogue contribution, the dialogue.
Mapping functions between ranks and rank-specific interpretative models
define the overall architecture of a linguistic theory.
Syntagmatic relation: A compositional relation, definable as
A fundamental distinction between (possibly universal) immediate dominance (ID) or part-whole relations and (partly language specific) linear precedence (LP) or temporally and spatially interpretable part-part relations is made in most computational grammars. For example, the ID structure of compound words in English and French is similar, but English is `right-headed' whereas French is `left-headed' and uses interfixed prepositions: peau-rouge `redskin', épingle à cheveux `hairpin', pain d'épice `gingerbread'.
In the ILEX approach, the core type of syntagmatic relation is the ID relation, and the LP relation is generalised to the quasi-linear precedence (QLP) relation in order to include prosodic association for suprasegmentals in speech, highlights and layout in writing. The QLP relation plays a similar role in surface form interpretation to logical form (LF) in semantic interpretation. A distinction is therefore made between compositional syntagmatic relations and interpretative syntagmatic relations; it is the latter which generally features in traditional descriptions. In current theories of syntax, syntagmatic relations are formalised as operations of compositionality, e.g. the slash and position operations in categorial grammar, rewrite and concatenation operations in phrase structure grammar, and the ID and LP relations of unification grammar.
A straightforward definition of a syntagmatic relation is as follows:
where at most one of x or y or z may remain uninstantiated, SynRel is a syntagmatic relation, and FS is a feature structure (i.e. AVM). For example,
This formula expresses Frege's Principle (FP)
(cf. [Cresswell 1973]) of compositionality,
i.e. the principle that a property of the whole is a function of this property
of the parts, whereby the function f may be concatenation, unification,
slash cancellation, etc., depending on the formalism used.
FP is generally applied only to
semantic interpretation; in the present approach it is also applied to
surface form interpretation. The function 
is less general in this
case than the surface interpretation functions, and needs components to
account for metaphor and ellipsis.
Paradigmatic relation:
A generalisation relation, characterising
similarity between signs in terms of one or more sign properties, defining
sets or classes,
elements of sets,
and set-subset inclusion, with the usual set theoretic operations
of union, intersection, and the formation of set theoretic relations as
tuples. Sign properties are defined in terms of feature structures, and
similarity is defined in terms of the subsumption (
) operation
.
Traditionally, paradigmatic relations
define semantic fields, syntactic categories, phonological natural classes,
and distributional classes of all kinds.
Leaving aside some technical details,
the terms used may be defined straightforwardly as follows, with
feature structures representing complex lexical properties of quantifiable
lexical objects, 
(the subsumer)
and 
(the subsumed)
are feature structures consisting of attribute-value (AV) pairs,
and `
' and `
' represent
conditional and biconditional propositional functions respectively:
| Subsumption: |
 |
| Paradigmatic relation: |
 |
Paradigmatic generalisations are expressed as inheritance relations between subclasses and classes, and among the subclasses of a given class. This concept is explained in the following sections.
