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Leonardi.DB
a logical geometry project

# Structures of Oppositions in Public Announcement Logic (2012), p. 335 by Demey, Lorenz

### Caption

Rhombic dodecahedron of oppositions for the epistemic properties of public announcement

### Logic

Aristotelian family
Classical Sigma-7
Boolean complexity
4
Number of labels per vertex (at most)
1
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

### Geometry

Shape
Rhombic Dodecahedron (irregular)
Colinearity range
0
Coplanarity range
0
Cospatiality range
0
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

### Style

Diagram is colored
No
Diagram is embellished
No
Tags
Boolean closed
;
Leuven

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. $[!\varphi]\neg K\psi \wedge (\varphi \vee K[!\varphi]\psi)$