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Leonardi.DB
a logical geometry project # How to Square Knowledge and Belief (2012), p. 310 by Lenzen, Wolfgang ### Caption

Doxastic square of opposition

### Logic

Aristotelian family
Lenzen Sigma-4
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
Rectangle (irregular)
Colinearity range
0
Coplanarity range
2
Cospatiality range
0
By central symmetry

### Graph structure

$B(a,p)$
• Implicit contradiction with $\neg B(a,p)$
• Implicit contrariety with $C(a,\neg p)$
• Implicit contrariety with $B(a,\neg p)$
• Implicit subcontrariety with $\neg C(a,p)$
• Subalternation from $C(a,p)$
• Implicit subalternation to $\neg C(a,\neg p)$
• Subalternation to $\neg B(a,\neg p)$
$C(a,p)$
• Implicit contradiction with $\neg C(a,p)$
• Implicit contrariety with $\neg B(a,p)$
• Implicit contrariety with $C(a,\neg p)$
• Implicit contrariety with $B(a,\neg p)$
• Subalternation to $B(a,p)$
• Implicit subalternation to $\neg C(a,\neg p)$
• Implicit subalternation to $\neg B(a,\neg p)$
$\neg C(a,\neg p)$
• Implicit contradiction with $C(a,\neg p)$
• Implicit subcontrariety with $B(a,\neg p)$
• Implicit subcontrariety with $\neg B(a,p)$
• Implicit subcontrariety with $\neg C(a,p)$
• Implicit subalternation from $C(a,p)$
• Implicit subalternation from $B(a,p)$
• Subalternation from $\neg B(a,\neg p)$
$\neg C(a,p)$
• Implicit contradiction with $C(a,p)$
• Implicit subcontrariety with $B(a,p)$
• Implicit subcontrariety with $\neg B(a,\neg p)$
• Implicit subcontrariety with $\neg C(a,\neg p)$
• Implicit subalternation from $C(a,\neg p)$
• Subalternation from $\neg B(a,p)$
• Implicit subalternation from $B(a,\neg p)$
$B(a,\neg p)$
• Implicit contradiction with $\neg B(a,\neg p)$
• Implicit contrariety with $B(a,p)$
• Implicit contrariety with $C(a,p)$
• Implicit subcontrariety with $\neg C(a,\neg p)$
• Subalternation from $C(a,\neg p)$
• Subalternation to $\neg B(a,p)$
• Implicit subalternation to $\neg C(a,p)$
$\neg B(a,p)$
• Implicit contradiction with $B(a,p)$
• Implicit contrariety with $C(a,p)$
• Implicit subcontrariety with $\neg B(a,\neg p)$
• Implicit subcontrariety with $\neg C(a,\neg p)$
• Subalternation from $B(a,\neg p)$
• Implicit subalternation from $C(a,\neg p)$
• Subalternation to $\neg C(a,p)$
$\neg B(a,\neg p)$
• Implicit contradiction with $B(a,\neg p)$
• Implicit contrariety with $C(a,\neg p)$
• Implicit subcontrariety with $\neg C(a,p)$
• Implicit subcontrariety with $\neg B(a,p)$
• Subalternation from $B(a,p)$
• Implicit subalternation from $C(a,p)$
• Subalternation to $\neg C(a,\neg p)$
$C(a,\neg p)$
• Implicit contradiction with $\neg C(a,\neg p)$
• Implicit contrariety with $C(a,p)$
• Implicit contrariety with $B(a,p)$
• Implicit contrariety with $\neg B(a,\neg p)$
• Implicit subalternation to $\neg B(a,p)$
• Implicit subalternation to $\neg C(a,p)$
• Subalternation to $B(a,\neg p)$

### 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

### Edge description

Contains definitions of relations
No
Form
solid lines
,
none
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