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Leonardi.DB
a logical geometry project

# Normatively Determined Propositions (2022), p. 84 by Pascucci, Matteo; Pizzi, Claudio

### Caption

Aristotelian KD-cube for the eight dyadic operators

### Logic

Aristotelian family
Lenzen Sigma-4
Boolean complexity
5
Number of labels per vertex (at most)
2
Equivalence between (some) labels of the same vertex
No
Analogy between (some) labels of the same vertex
No
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

### Geometry

Shape
Cube (regular)
Colinearity range
0
Coplanarity range
0
Cospatiality range
0
By some other geometric feature

### Vertex description

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

### Edge description

Contains definitions of relations
No
Form
dotted lines
,
solid lines
,
none
,
dashed lines
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

The two sequences of subalternations in this Lenzen sigma-4 look as follows:

$\triangle^*(A,B) \to \triangle(A,B) \to \blacktriangle(A,B) \to \blacktriangle^*(A,B)$

$\triangledown^*(A,B) \to \triangledown(A,B) \to \blacktriangledown(A,B) \to \blacktriangledown^*(A,B)$

P. 81:

$\triangle(A,B) := \Diamond A \wedge (\Box(A \to B) \vee \Box(A \to \neg B))$, meaning that $A$ is permitted and $B$ is normatively determined by $A$

$\triangle^*(A,B) := \Box A \wedge (\Box(A \to B) \vee \Box(A \to \neg B))$, meaning that $A$ is obligatory and $B$ is normatively determined by $A$

$\blacktriangle(A,B) := \Diamond A \to (\Box(A \to B) \vee \Box(A \to \neg B))$, meaning that if $A$ is permitted, then $B$ is normatively determined by $A$

$\blacktriangle^*(A,B) := \Box A \to (\Box(A \to B) \vee \Box(A \to \neg B))$, meaning that if $A$ is obligatory, then $B$ is normatively determined by $A$

$\triangledown(A,B) := \Diamond A \wedge (\Diamond(A \wedge B) \wedge \Diamond(A \wedge \neg B))$, meaning that $A$ is permitted and $B$ is not normatively determined by $A$

$\triangledown^*(A,B) := \Box A \wedge (\Diamond(A \wedge B) \wedge \Diamond(A \wedge \neg B))$, meaning that $A$ is obligatory and $B$ is not normatively determined by $A$

$\blacktriangledown(A,B) := \Diamond A \to (\Diamond(A \wedge B) \wedge \Diamond(A \wedge \neg B))$, meaning that if $A$ is permitted, then $B$ is not normatively determined by $A$

$\blacktriangledown^*(A,B) := \Box A \to (\Diamond(A \wedge B) \wedge \Diamond(A \wedge \neg B))$, meaning that if $A$ is obligatory, then $B$ is not normatively determined by $A$