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Leonardi.DB
a logical geometry project

Logik im Recht. Grundlagen und Anwendungsbeispiele (2010), p. 213
by Joerden, Jan C.

Caption

Die Beziehungen zwischen Sollen und Sein (ergänzte Version)

Legend

Logic

Aristotelian family
Sigma-6 Graph
Boolean complexity
5
Number of labels per vertex (at most)
1
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

Geometry

Shape
Hexagon (regular)
Colinearity range
0
Coplanarity range
6
Cospatiality range
0
Representation of contradiction
By central symmetry

Vertex description

Conceptual info
No
Mnemonic support (AEIO, purpurea ...)
No
Form
dots
Label type
symbolic
Symbolic field
logic
Contains partial formulas or symbols
Yes
Logical system
deontic logic
,
modal logic

Edge description

Contains definitions of relations
No
Form
dotted lines
,
solid lines
,
none
,
dashed lines
Has arrowheads
Yes
Overlap
No
Curved
No
Hooked
No
As wide as vertices
No
Contains text
No
Label type
none

Style

Diagram is colored
No
Diagram is embellished
No

Additional notes

The innermost hexagon consists of the alethic modalities; N = $\Box p$, U = $\Box\neg p$, K = $\Diamond p\wedge\Diamond\neg p$.
The outermost hexagon consists of the deontic modalities; G = $O p$, V = $O\neg p$, I = $P p\wedge P\neg p$.

This diagram induces the following 5-partition:
1) N ($\Box p$)
2) G $\wedge$ K = G $\wedge$ $\neg$N ($Op \wedge \Diamond\neg p$)
3) I ($Pp \wedge P\neg p$)
4) V $\wedge$ K = V $\wedge$ $\neg$U ($O\neg p \wedge \Diamond p$)
5) U ($\Box \neg p$)
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