Bishop Gregory (hgr) wrote,
Bishop Gregory

модальности (9)

про дистанционную модальность (логика предпочтений применительно к пространству).1 Distance space modality

Presently, there are many logical systems elaborated for reasoning about (different kinds of) space. Normally, all of them consist from two components corresponding to the two mutually irreducible types of quantifiers (‹1, 1› and ‹1, 1, 1›): a kind of the topological modal logic (quantifiers of ‹1, 1› type) and logic of distances in relative coordinates (quantifiers of ‹1, 1, 1› type, often called “ternary” by spatial logicians). The latter is governed by such operators as “nearer than” (van Benthem 1983; van Benthem, Bezhanishvili 2007, 406), “closer-than” (van Benthem, Bezhanishvili 2007, 394-395) or “closer” (Sheremet, Tishkovsky, Wolter, Zakharyaschev, 2005), etc.

Von Wright’s (1979) “logic of elsewhere” (for its recent developments see (van Benthem, Bezhanishvili 2007, 405)) with its main operators “elsewhere” and “everywhere” is especially interesting from the philosophical point of view as an example of clear balancing between the two modal approaches. Indeed, the modal approach to distance presupposes that the observer is involved, but this fact is often implicit and far from being evident. But such categories as “here” and “elsewhere” make it explicit.

The “ternary” operators of the spatial logic are nothing but the main operator of the logic of preferences more than. Indeed, von Wright’s “logic of elsewhere” and, especially, the logic of “closer” (Sheremet, Tishkovsky, Wolter, Zakharyaschev, 2005) are not specifically spatial by nature and could be easily reshaped for other modalities.

Thus, the basic principles of the logic of preferences are applicable to the spatial logic of distances.

To sum up, the logic of space is twofold in the same manner as are alethic, deontic, axiological, and epistemic modal reasoning. One part of it deals with topology, that is, the structure of the modal states of the space. Another part of it deals with distances, that is, the preferences in space.


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