Relativism and Universalism in Linguistics - Fachbereich 10 ...

110 Sections
We distinguish sense, meaning and reference. The term meaning of some fragment of a given
text is accepted as the content grasped in some particular situation of reading following the
reader’s presuppositions and prejudices in the interpretative process, formalized by the term
sense. For the text understanding is not postponed until the final sentence, it should have
meaningful parts; their meanings determine the meaning of the whole. We argue that:
i. an arbitrary union of meaningful parts of an admissible text is meaningful;
ii. a nonempty intersection of two meaningful parts of an admissible text is meaningful.
For an admissible text X is meaningful by definition, it remains to define formally the
meaning of its empty part in order to provide X with some topology in a strict mathematical
sense, where open sets U ⊂ X are all its meaningful parts. The ordinary reading process
inherits a natural temporality of phonetic phenomena, that’s why we call phonocentric the
topology so defined. For any two different sentences x, y of an admissible text X, there is an
open U ⊂ X containing precisely one of them; whence the phonocentric topology should
satisfy the separation axiom T0 of Kolmogoroff. An admissible text X gives rise to a finite
space, hence it is an A-space. For a sentence x ∈ X, we define Ux to be the intersection of all
the meaningful parts that contain x, i.e.the smallest open neighborhood of x. We define the
specialization relation ≼ on X by setting x ≼ y if and only if x ∈ Uy or, equivalently, Ux ⊆ Uy.
Note that for all x, y ∈ X, x ≼ y implies x ≤ y, where ≤ defines the usual linear order of
reading.
Proposition. The set of all open sets of the kind Ux is a basis of a phonocentric topology on
X. Moreover, it is the unique minimal basis of a phonocentric topology. The phonocentric
topology on an admissible text defines a partial order ≼ on it by means of specialization; the
initial phonocentric topology can be recovered from this partial order ≼ in a unique way.
Note that another concept of meaning or criteria of meaningfulness would imply another
definition of meaningful fragments and so will define yet another type of semantic topology
which may be thought of as a kind of linguistic universal because it is invariant under
translation.
There exists a simple intuitive tool called Hasse diagram for the graphical representation of a
finite partially ordered set (poset). For a poset (X, ≼), the cover relation ≺ is defined by:
‘x ≺ y if and only if x ≼ y and there exists no element z∈X such that x≼z≼y’. For a given poset
(X,≼), its Hasse diagram is defined as the graph whose vertices are the elements of X and
whose edges are those pairs {x, y} for which x≺y. The usage of some kind of Hasse diagram
under the name Leitfaden is widely spread in mathematical treatises. In this usage, the poset is
constituted of all chapters of the book. The Hasse diagram of this kind may be “split” in order
to draw the Hasse diagram whose vertices are all paragraphs and then, all sentences; this gives
the graphical representation of a phonocentric topology at the semantic level of text.
These considerations may be modified in order to define a phonocentric topology at the
semantic level of sentence. Thus for a given admissible text, we can find, in a constructive
manner, its phonocentric topology at each semantic level. Likewise for any other type of
semantic topology. Now we may interpret linguistic notions in terms of topology and order
and undertake their geometric studies, which may be thought of as a kind of universal syntax.
References: My works on formal linguistics, and standard sources in relevant mathematics.