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Leonardi.DB
a logical geometry project

# A Cube of Opposition for Predicate Logic (2020), p. 106 by Nilsson, Jørgen Fischer

### Caption

Contradictions, complementations, quantifier inversions (color figure online)

### Logic

Aristotelian family
Buridan Sigma-4
Boolean complexity
6
Number of labels per vertex (at most)
2
Equivalence between (some) labels of the same vertex
Yes
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
Label type
symbolic
Symbolic field
logic
Contains partial formulas or symbols
Yes
Logical system
predicate logic

### Style

Diagram is colored
Yes
Diagram is embellished
No
Tags
composed operator duality

Let $C,D$ be unary predicates and $R$ a binary relation.

$\forall\forall$ stands for $\forall x( Cx \to \forall y (Dy \to Rxy)$
$\forall\exists$ stands for $\forall x( Cx \to \exists y (Dy \wedge Rxy)$
$\exists\forall$ stands for $\exists x( Cx \wedge \forall y (Dy \to Rxy)$
$\exists\exists$ stands for $\exists x( Cx \wedge \exists y (Dy \wedge Rxy)$

$\forall\forall\neg$ stands for $\forall x( Cx \to \forall y (Dy \to \neg Rxy)$
$\neg\exists\exists$ stands for $\neg\exists x( Cx \wedge \exists y (Dy \wedge Rxy)$

$\forall\exists\neg$ stands for $\forall x( Cx \to \exists y (Dy \wedge \neg Rxy)$
$\neg\exists\forall$ stands for $\neg\exists x( Cx \wedge \forall y (Dy \to Rxy)$

$\exists\forall\neg$ stands for $\exists x( Cx \wedge \forall y (Dy \to\neg Rxy)$
$\neg\forall\exists$ stands for $\neg\forall x( Cx \to \exists y (Dy \wedge Rxy)$

$\exists\exists\neg$ stands for $\exists x( Cx \wedge \exists y (Dy \wedge\neg Rxy)$
$\neg\forall\forall$ stands for $\neg\forall x( Cx \to \forall y (Dy \to Rxy)$

(Cf. pp. 104-105.)