Bishop Gregory (hgr) wrote,
Bishop Gregory
hgr

модальности (7)

о пространственных вообще и о топологических в частности.1 ­Modality of space

Space and time are never considered as modalities in linguistics, but the only probable explanation of this fact is little knowledge of the corresponding logical studies by the linguists. In the narratology, the situation is almost the same but Vadim Rudnev (2000) proposed, while without any appropriate logical apparatus, to consider space and time among the modalities which are shaping the fictional plot. I think his intuition is right.

Let us start from the spatial modality because it is the field where we have an about 80-year scholarly tradition to rely on (see now especially (Aiello, Pratt-Hartmann, van Benthem, 2007) with further bibliography).

The modal logic of space was born at the turn of the 1930s and 1940s, when several scholars independently (Stone in 1937, Tarski in 1938, Tsao-Chen in 1938, and McKinsey in 1941) discovered that the most known now (and quite recently developed then) modal logic of predicates, S4, could be interpreted as a logic of topological space (Kontchakov, Kurucz, Wolter, Zakharyaschev 2007, 509). The topological space is distinct from the space(s) of human perceptions but now we need only a rough scheme and so, even topological spaces could provide a satisfying approximation. To be closer to reality, we would need to operate with something like non-metric spaces on the fuzzy graphs, but, even in this case, we will need to construct basic notions analogous to those defined for the topological spaces.

There are two kinds of modal logics of space. The first one (developed first and mentioned above) deals with the topological properties (in a larger sense of word, not restricted to the so-called topological spaces) and, using our previous terminology, is governed by the ‹1, 1› type quantifiers. The second one deals with the distances and is governed by the ‹1, 1, 1› type quantifiers.

2 Modalities of topology

It is necessary to start from, at least, basic definitions (basing on (Kontchakov, Kurucz, Wolter, Zakharyaschev 2007) with minor additions on the graphs).

Space: a pair (Δ, d) where Δ is a nonempty set (of points or vertices of graphs or vertices of fuzzy graphs) and d is a function from Δ × Δ into a set of non-negative real numbers ³0 (or, for spaces on graphs, its subset [set of natural numbers]), satisfying, at least, the first from the three following axioms:

(1) d (x, y) = 0 iff x = y (identity of indiscernibles),

(2) d (x, y) = d (y, x) (symmetry),

(3) d (x, z) £ d (x, y) + d (y, z) (triangle inequality),

for all x, y, z Δ.

The value d(x, y) is called the distance between the elements x and y. If these elements are not points but vertices of graph, the distance is defined by the number of the vertices in between, as it is usual in the graph theory.

The metric space is defined on the set of points, its d-function’s range of values is ³0, and it satisfies the whole set of the three axioms.

The distant space is similar to the metric space but does not satisfy the axioms (2) and (3). (On the modal logic for such spaces see, especially (Kurucz, Wolter, Zakharyaschev, 2005).

Examples of breaking condition (2). It is easy to see how, in human perception, the way forward and the way back could differ in length. Not only in human perception, though. The beloved example of Kurucz, Wolter, and Zakharyaschev is the flight from London to Tokyo and back. London is “farer” from Tokyo than Tokyo from London by more than one hour.

Examples of breaking condition (3). The distances are often evaluated in fuzzy terms, such as “short,” “long,” and so on. Short plus short can still be short, but it can also be medium or long. (Here Kurucz–Walter–Zakharyaschev’s example is, in fact, a space on a fuzzy graph).

Main topological operators are those of interior I and closure C. (I omit here the strict definition of the topological space and its interior by means of the four Kuratowski axioms.) C is dual of I. If U is a nonempty set, the universe of the space, and X U, then, CX = U – I(UX). A set X is called open if X = IX and closed if X = CX. The complement of an open set is closed and vice versa. The boundary of X is CX – IX. The self-evident meaning of these definitions is that a set X of the universe of the space U can be considered as either including its boundary (operator C) or excluding it (operator I).

Topological operators for the metric spaces. The main sense of the following definitions consists in the fact that the presence or the absence of the boundary of a set X of the metric space (Δ, d) affects drastically the localization of some element (point) y vis-à-vis an element (point) x X. If X is open, then, there is a way to formulate a condition that any y X, too. If X is closed, then, there is a way to formulate a condition that only some y X. Thus, the interior operator becomes an equivalent of the modal necessity, and the closure operator that of the modal possibility.

The definitions are the following:

IX = {x X | ε > 0 y (d(x, y) < ε y X)},

CX = {x Δ | ε > 0 y X d(x, y) < ε}.

These equations can be reformulated in simple words as following: IX is a set whose all elements are internal, while CX is a set whose not all elements are internal but some of them belong to the boundary. Such formulations make self-evident why the operator I is equivalent to the modal operator of necessity () while the operator C to the operator of possibility ().

For the spaces on the graphs, the notion of ε-vicinity has no sense. However, an analogous definition of the notion of closure is working. Its meaning is to provide the maximum number of links (edges) between the vertices of a given graph. Thus,

· the closure of the graph G having n vertices is a graph C(G) obtained from G by recursively joining pairs of non-adjacent vertices u and v, for which degree (u) + degree (v) ≥ n until no such pair remains (where “degree” of a vertex is the number of the edges adjacent to that vertex).

It follows from the definition of C(G) that I(G) = G (cf. definition of the open set in the topological space. Two graphs C(G) and G differ from each other by the edges: the edges which they have in common are “internal” and the edges proper to C(G) are, so-to-say, “boundary” of the graph. Thus, G (more properly, I(G)) is an equivalent of logical necessity (): it represents a space where any existing path (edge) is internal. C(G) is an equivalent of logical possibility () given that only some its existing paths (edges) are internal, that is, belonging to G.

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