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Leonardi.DB
a logical geometry project # A Cube of Opposition for Predicate Logic (2020), p. 107 by Nilsson, Jørgen Fischer ### Caption

Cube of opposition with R being equality

### Logic

Aristotelian family
Buridan Sigma-4
Boolean complexity
5
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
Form
none
Label type
linguistic
Language
English
Lexical field
unusual construction
Contains partial sentences or single words
No
Contains abbreviations
Yes

### Style

Diagram is colored
No
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.)

Note that interpreting R as identity does not change the Aristotelian family of the diagram (it remains a Buridan sigma-4), but it does imply that the Boolean complexity of this diagram goes down from 6 to 5. Cf. Lorenz Demey, 2019, Boolean considerations on John Buridan's octagons of opposition, History and Philosophy of Logic, 40: 116-134 (link).