Invariants of Natural Language
Edward L. Keenan and Edward Stabler
Department of Linguistics,
UCLA, Box 951543
Los Angeles, CA 90095-1543
(ekeenan@ucla.edu)
Prerequisites: Some familiarity working with functions and relations.
Summary: Lecture 1: Bare Grammar (based on joint work with Ed Stabler).
We offer a fully mathematical and notation neutral conception of generative grammar. In
terms of this we define two, independent, notions: syntactic invariant and
semantic invariant. We focus on the former and provide a notation neutral
characterization of grammatical constant ("function word"), grammatical
property and grammatical relation. The characterization is done in
terms of invariance under syntactic automorphisms (bijections from the language
to the language which preserve how expressions are derived from the
"lexicon").
Lecture 2: We illustrate syntactic invariants with several
"mini-grammars". E.g. in our model of Korean the case markers are (as expected) grammatical
constants. Equally the Anaphor-Antecedent relation is invariant,
defined in terms of morphological identity of case markers, not in terms
of hierarchical structure, such as C-command. We offer as a language
universal that the Anaphor-Antecedent relation is always a structural
invariant of a grammar, but not uniformly definable. By contrast
entailment is not invariant.
Lecture 3: Universal Invariants We show that (1) a variety of
mathematically natural operations (boolean ones for example) preserve and
determine invariants in a way that is useful (but expected); equally we show
that several specifically linguistic relations, such as is a constituent
of, C-commands (suitably generalized), and is a cycle of length n (a
notion we define), are invariant in all grammars.
Lecture 4: A Mathematical Theory of Grammar Categories We study several
axioms that constrain the form of possible human languages using,
crucially, the notion of grammatical category. Specifically:
- A1: Substitutivity: Recursively intersubstitutable expressions
have the same category (the converse in argued not to be valid
for natural language grammars).
- A2: Stability: For each category C, the property of being a phrase
of category C is a stable invariant (properties, relations,...) preserved by stable
automorphisms, ones that extend to extensions of the language in certain ways.
A2 allows that structure maps change category, but it requires that they do so uniformly:
whenever s and t have the same category then so do their images under an automorphism.
- A4: Recursion: Every grammar has a proper cycle. The notion of
cycle is defined and is a starting point for characterizing the
types of recursion in natural language. (It is clearly not
sufficient as given at the moment).
A Universal characterization of category types: Modifier
Categories, Predicate Categories, Argument Categories are characterized in
universal terms.
Lecture 5: Semantic Correlates of Syntactic Universals. We offer a
variety of empirically observed correlates, both syntactic and semantic, of
invariants at the expression, property and relation levels. These do not
(at time of writing) follow from our notion of invariant.
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