Structures of Oppositions in Public Announcement Logic (2012), p. 336
by Demey, Lorenz

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Caption

Six hexagons that are embeddable in the rhombic dodecahedron

Logic

Aristotelian family

Jacoby-Sesmat-Blanché Sigma-3

Boolean complexity

3

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 (irregular)

Colinearity range

0

Coplanarity range

0

Cospatiality range

0

Representation of contradiction

By central symmetry

Vertex description

Conceptual info

No

Mnemonic support (AEIO, purpurea ...)

No

Form

none

Label type

symbolic

Symbolic field

logic

Contains partial formulas or symbols

No

Logical system

epistemic logic

,

dynamic logic

Edge description

Style

Diagram is colored

No

Diagram is embellished

No

Tags

Boolean closed

;

Leuven

Additional notes

The numbers 1-14 stand for the following formulas from public announcement logic (cf. p. 332):

1. $\varphi$
2. $\langle!\varphi\rangle K\psi$
3. $K[!\varphi]\psi$
4. $[!\varphi]K\psi$
5. $\neg\varphi$
6. $\langle!\varphi\rangle\neg K\psi$
7. $\neg K[!\varphi]\psi$
8. $[!\varphi]\neg K\psi$
9. $\neg\varphi \wedge K[!\varphi]\psi$
10. $\neg\varphi \wedge \neg K[!\varphi]\psi$
11. $\varphi \vee \neg K[!\varphi]\psi$
12. $\varphi \vee K[!\varphi]\psi$
13. $[!\varphi]K\psi \wedge (\varphi \vee \neg K[!\varphi]\psi)$
14. $[!\varphi]\neg K\psi \wedge (\varphi \vee K[!\varphi]\psi)$