Exkurs:Mereogeometries

Exkurs zu: Raum und Geometrie

Mereogeometries are a group of particular logical formalisms for describing n-dimensional space that are based on mereological calculi. Mereology’s focus of interest is part-whole relations. For instance, Clarke’s mereological calculus ([Clarke 1981a]Clarke, Bowman, L. (1981).
A Calculus of Individuals Based on ‘Connection’. In Notre Dame Journal of Formal Logic, 22, 3, 204-218.

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; here quoted from [Vieu 1991a]Vieu, Laure (1991).
Séman­tique des rela­tions spatia­les et infé­rences spatio-tempo­relles: Une contri­bution á l'étude des structures formelles de l'espace en Langage Natu­rel. Tou­louse: INRI, Uni­versi­té Paul Saba­tier, Doctoral Thesis.

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: S. 120ff) is based on the primitive relation C(x, y) with the intended meaning “individual x is connected with individual y” and defined by the axioms:
 A0.1 ∀ x (C(x, x)) ∧ ∀ x ∀ y (C(x, y) → C(y, x)) Axiom of reflexivity and symmetry A0.2 ∀ x (∀ y ∀ z (C(z, x) ↔ C(z, y)) → x=y) Axiom of extension

Some definitions possible in Clarke’s calculus are …

 D0.1 DC(x, y) ≡def ᓓ C(x, y) : “x is disconnected with y” D0.2 P(x, y) ≡def ∀ z (C(z, x) → C(z, y)) “x is part of y“ D0.3 PP(x, y) ≡def P(x, y) ∧ ᓓ P(y, x) “x is a proper part of y” D0.4 O(x, y) ≡def ∃ z (P(z, x) ∧ P(z, y) “x overlaps y“ D0.6 EC(x, y) ≡def C(x, y) ∧ ᓓ O(x, y) “x is externally connected with y” D0.7 TP(x, y) ≡def P(x,y) ∧ ∃ z (EC(z, x) ∧ EC(z, y)) “x is a tangential part of y D1.1 x=F(α) ≡def ∀ y (C(y, x) ↔ ∃ z (z∊α ∧ C(y, z)) “x is identical to the fusion of the set of individuals α” D1.2 x+y ≡def F({z : P(z,x) ∨ P(z, y)}) “x+y is the sum of x and y” D1.5 x∧y ≡def F({z : P(z,x) ∧ P(z, y)}) “x∧y is the intersection of x and y” Illustrations to the relations O, TP, and EC

… so that, for instance, the following theorem can be proven:

 T0.34 ∀ x ∀ y ∀ z ((TP(z, x) ∧ P(z, y) ∧ P(y, x)) → TP(z, y)

A definition of “point“ out of a set α of individuals (with Λ being the empty set):

 PT(α) ≡def ᓓ α=Λ ∧ ∀ x ∀ y ((x∊α ∧ y∊α) → (EC(x, y) ∨ (O(x, y) ∧ x∧y∊α))) ∧ ∀ x ∀ y ((x∊α ∧ P(x, y)) → y∊α) ∧ ∀ x ∀ y (x+y∊α → (x∊α ∨ y∊α)) Datei:Mereogeom2.jpg Illustration to the construction of points

(i.e., all individuals partaking in a point are connected with each other; if two of them overlap, their intersection is also part of the point; each individual containing an element of the point is also element of that point; and finally, if an element of the point is the sum of two individuals then those must be elements as well.)

This calculus already allows dealing with topological relations and can be extended easily to a full geometry (i.e., including directions and metric distance). That is, any geometrical configuration can be described by a set of propositions of that calculus. Any analysis or transformation of the geometrical configuration can correspondingly be performed in analogy with the set of propositions by means of logical analyses or transformations (cf., e.g., [Pratt-Hartmann 2000a]Pratt-Hartmann, Ian (2000).
Empi­ricism and Ratio­nalism in Re­gion-based Theories of Space. In Funda­menta Infor­maticae, 34, 1–31.

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).

Anmerkungen
Literatur                             [Sammlung]

[Clarke 1981a]: Clarke, Bowman, L. (1981). A Calculus of Individuals Based on ‘Connection’. Notre Dame Journal of Formal Logic, Band: 22, Nummer: 3, S. 204-218.

[Pratt-Hartmann 2000a]: Pratt-Hartmann, Ian (2000). Empi­ricism and Ratio­nalism in Re­gion-based Theories of Space. Funda­menta Infor­maticae, Band: 34, S. 1–31. [Vieu 1991a]: Vieu, Laure (1991). Séman­tique des rela­tions spatia­les et infé­rences spatio-tempo­relles: Une contri­bution á l'étude des structures formelles de l'espace en Langage Natu­rel. Tou­louse: INRI, Uni­versi­té Paul Saba­tier, Doctoral Thesis.

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